I’ve added a ♬ next to books I think have some cultural content. This is pretty subjective, of course.
I’ve added a next to books I particularly liked.

きこえる？きこえるよ
Can you hear? I can hear! No words. Use to see if Ehon navi works for you!

レインボウブックス ながれ
Flow, no words

レインボウブックス きもち
Feelings, no words

COLORS (カラーズ)
Colors, 11 words

するするすとーん
Slide, Slide, Plunk, 15 words

もこもこもこ
Bulge Bulge, 16 words

レインボウブックス ことば
Words, 16 words

ぐるぐるどぼーん
Spin, Spin, Plop, 16 words

GREETINGS（グリーティングス）あいさつのえほん
Picture Book of Greetings, 17 words

いっしょにあそぼう おめめ おはな おくち
Let’s Play! Eyes, Nose, Mouth, 17 words

おはよう
Good Morning, 18 words

いただきまーす！
Let’s Eat! 18 words

のれるかな？ うかぶかな？ ききゅう
Balloon, 20 words

ぽんぽんポコポコ
Thump thump, 20 words

ころころぽーん
Roll, Roll, Bounce, 20 words

いっしょにあそぼう あいうえお
Let’s Play! A-I-U-E-O, 20 words

おでこ ぴたっ
Forehead Bump, 22 words

おてて たっち
Touch Hands, 23 words

いない いない おかお
Where’s That Face? Peekaboo!, 25 words

もっと いろいろ ばあ
More Color Peekaboo, 25 words

おはな つんつん
Nose Kisses, 25 words

めくってごらん
Let’s Turn The Page, 25 words ♬

つみき
Building Blocks, 27 words

いろいろばあ
Color Peekaboo, 30 words

に～っこり
Big Smile, 30 words

しーっ
Shhh, 30 words

だっだぁー
Daddaa, 30 words

レインボウブックス リズム
Rhythm, 30 words

プララのとんねるぶっぶー
Purala’s Tunnel *honk honk*, 30 words

さがしてごらん
Let’s Search, 32 words

おやすみ ペンギン
Good Night, Penguin, 35 words

かんぱーい
Cheers, 35 words

ノンタンはみがきはーみー
Nontan Brushes His Teeth, 35 words ♬

どんぐりころちゃん
The Little Roly-Poly Acorn, 35 words

ぶっぶー　どらいぶ
*honk honk* Driving, 40 words

みて みて おてて
Look, Look, My Hand, 40 words

赤ちゃんのための色のえほん
Color Picture Book for Babies, 40 words

赤ちゃんのためのことばの絵本
Word Picture Book for Babies, 40 words

おでんわ もしもし しましまちゃん
It’s for you! Hello, Stripe-Chan, 40 words ♬

アリのおでかけ
The Ants Go Out, 40 words

ねんねのうた
The Night-Night Song, 40 words

あるひ、こねこね
One Day, *knead knead*, 45 words

なに なに なあに？
What’s This, What’s This, What’s Thiiiis?, 45 words

おさんぽ さんぽ
A Walk, A Walk, 45 words

はなび ドーン
Fireworks! Boom!, 45 words

ありがとう
Thank You, 45 words

のぞいてごらん
Let’s Peek, 45 words

れんけつガッチャン
Connect! *click*, 45 words

ぼくのいす？
Is This My Chair?, 50 words

赤ちゃんのためのかたちの絵本
Picture Book of Shapes for Babies, 50 words

くるくる なあに？
What’s Rolled Up?, 50 words

ころころとんとん
Roll, Roll, Bounce, Bounce, 50 words

まねっこ にゃー にゃー
Copy Cat Meow Meow, 50 words

くらべてごらん
Let’s Compare, 50 words ♬

さわさわもみじ
Rustling Maple Leaves, 50 words

おもちゃのかくれんぼ
The Toys Play Hide and Seek, 55 words

ありさんぽつぽつ
The Ants Go Marching, 55 words

ゆき ふふふ
The Snow Falls, 55 words

ほわほわさくら
Fluttering Cherry Blossoms, 55 words

あめ ぽぽぽ
Raindrops Splash, 55 words

まゆげちゃん
Dad’s Eyebrows, 60 words

しろくまくんの このあな なぁに？
Polar Bear-kun, What’s This Hole?, 60 words

あかちゃんたいそう
Baby Exercises, 60 words

たどってごらん
Let’s Follow, 60 words

ぎょうれつ
The Line, 60 words

ねえ、あそぼ！ 2さいのパンダちゃん
Come On, Let’s Play! 2-year old Little Panda, 60 words

いろいろバス
Various Bus Colors, 65 words

きょうのそらはどんなそら
What’s today’s sky like? 70 words

おばけのアイスクリームやさん
The Ice-Cream Selling Ghost, 70 words

まってるまってる
Waiting, Waiting, 70 words

ほっぺに、ちゅー。
A Kiss On The Cheek, 70 words

くんくん、いいにおい
*sniff sniff* That Smells Good,72 words

うみ、ざざざ
The Sound of the Ocean, 75 words

かくれんぼ しましょ
Let’s Play Hide and Seek, 75 words

ほらそっくり
Look, We’re The Same, 75 words

とーんだ とんだ
Blown Away, 75 words

青いヤドカリ
The Blue Hermit Crab, 80 words

とんとんとん
Knock Knock Knock, 80 words

りんご　りんご　りんご　りんご　りんご　りんご
Apple Apple Apple Apple Apple Apple, 80 words

いろいろいちご
Many Kinds of Strawberries, 80 words

ココロのヒカリ
Soul Light, 80 words

赤ちゃんずかん（0・1・2さい）
Baby’s Illustrated Dictionary (For 0, 1 and 2 year olds), 80 words

スージー・ズー わらってるのはだあれ？
Suzy’s Zoo: Who’s Laughing?, 80 words

ないた
I Cried, 90 words ♬

おたんじょうびのケーキちゃん
The Little Birthday Cake, 90 words

おとうさん もういっかい ゆうえんち
One More, Papa! Papa’s Amusement Park, 90 words

おとうさん もういっかい うみあそび
One More, Papa! Papa’s Beach Fun, 90 words

おとうさん もういっかい たかいたかい
One More, Papa! Up High With Papa, 90 words

はたらく　じどうしゃ
Vehicles At Work, 100 words

ものすごくおおきなプリンのうえで
On Top Of A Huge Pudding, 100 words
(Some text was cut off when this got scanned, unfortunately.)

ねんねしよ。
Let’s Go Night-Night, 100 words

かわいいね。
Cute, Huh? 100 Words

スージー・ズー はみがきだいすき
Suzy’s Zoo: I Love To Brush My Teeth, 100 words

スージー・ズー いやいやウィッツィー
Suzy’s Zoo: Witzy Says No, No, 100 words

スージー・ズー おふろだいすき
Suzy’s Zoo: We Love Our Bath!, 100 words

スージー・ズー おやすみウィッツィー
Suzy’s Zoo: Goodnight, Witzy, 100 words

スージー・ズー あめがふってきたよ！
Suzy’s Zoo: It’s Raining!, 100 words

スージー・ズー エリーファントのティーパーティー
Suzy’s Zoo: The Elephant’s Tea Party, 100 words

スージー・ズー ウィッツィーのおはなばたけ
Suzy’s Zoo: Witzy’s Flower Garden, 100 words

スージー・ズー あきを あつめたよ！
Suzy’s Zoo: We Gathered Fall!, 100 words

スージー・ズー もうすぐハロウィーン
Suzy’s Zoo: Halloween Is Soon!, 100 words

スージー・ズー いつまでもともだち
Suzy’s Zoo: Forever Friends, 100 words

スージー・ズー ふゆってたのしいね！
Winter Is Fun, Isn’t It?, 100 words

いろいろサンドイッチ
Various Sandwiches, 100 words

つきよのかいじゅう
Monster in the Moonlight, 120 words

すいかのめいさんち
The Watermelon Town, 120 words ♬

ぼくんちカレーライス
We’re Having Curry Rice, 125 words ♬

ノンタン　いもうと　いいな
Nontan’s Sister’s Great, 125 words

ノンタンがんばるもん
Nontan’s Gonna Try Hard, 125 words

こけこっこー
Cock-A-Doodle-Do, 125 words

おいしいおほしさま
Delicious Stars, 125 words ♬

ぴよちゃんのおともだち
Pyo-chan’s Friend, 125 words

いち、にの、さーん。
One, Two, Three, Go!, 125 words

ノンタンスプーンたんたんたん
Nontan’s Spoon, 150 words

ないしょでんしゃ
Mystery Train, 150 words

あいうえおべんとう
A-I-U-E-O Lunchbox, 150 words ♬

おかあさんだもの
Because I’m Your Mother, 150 words

ともだち くろくま
Black Bear’s Friend, 150 words

むしコレ
Bug Collection, 150 words

すごいくるま
The Amazing Car, 150 words

にんげんごみばこ
The Garbage Can For Humans, 175 words

けいちゃんのくすりゆび
Kei-chan’s Medicine Finger, 175 words

くいしんぼなめれおん
Lickelon the Glutton, 175 words

ありさんとぞうさんのおさんぽ
Mr. Ant and Mr. Elephant Take A Walk, 175 words

大阪うまいもんのうた
The Song of Yummy Osaka Food, 175 words ♬
(This video will put this book in context.)

クリスマス くろくま
Black Bear’s Christmas, 175 words

ぴたっとヤモちゃん
Clingy Yamo-chan, 175 words

なめれおん
Lickelon, 200 words

サンタのおまじない
Santa’s Magic Words, 200 words

あいうえおのえほん
A-I-U-E-O Picture Book, 200 words

◯×うさぎ
Good-Bad Bunny, 200 words

とこやにいったライオン
The Lion That Went To The Barber, 200 words

おしろとおくろ
The Black Castle and the White Castle, 200 words ♬

あかにんじゃ
The Red Ninja, 200 words ♬

おばけのくに
The Ghost Country, 200 words

たからじまの ぼうけん
Treasure Island Adventure, 200 words

おたんじょうび くろくま
Happy Birthday, Black Bear, 200 words ♬

りんごちゃん
Little Miss Apple, 200 words

スージー・ズー あひるのウィッツィー
Suzy’s Zoo: Witzy the Duck, 200 words

おせちのおしょうがつ
Osechi for New Year’s, 225 words ♬

森のイスくん
Mr. Chair In The Forest, 250 words

おやすみ、はたらくくるまたち
Goodnight, Construction Equipment, 250 words

ハローキティはじめてのこどもずかん
Hello Kitty’s Baby’s First Illustrated Dictionary, 250 words

妖怪食堂
The Youkai Cafeteria, 250 words ♬

かっぱのあいさつ
Kappa Greetings, 275 words ♬

コドモザウルス
Kidasaurus, 275 words

くものこくーのかぞくでおでかけ
Cloud Boy Kuu Shops With His Parents, 275 words

もうママったら！
Jeez, Mama!, 275 words

あたし、ようせいにあいたい！
I want to meet a fairy!, 275 words

ダットさん
Mr. Datsun, 275 words

たなばたバス
Tanabata Bus, 275 words ♬

たかこ
Takako, 300 words ♬

もっちゃうもっちゃうもうもっちゃう
Can’t Hold It Anymore!, 300 words

3ぷんもまてないよ！
I Can’t Wait Three Whole Minutes!, 300 words

ゆめたまご
Dream Egg, 300 words

だんろのまえで
In Front of the Fireplace, 300 words

おつきみどろぼう
A Thief on Moon-Viewing Night, 300 words ♬

ボクのかしこいパンツくん
The Wise Mr. Underwear, 300 words

かめだらけおうこく
A Land Filled With Turtles, 350 words

くまくんのたまご
The Bear’s Egg, 350 words

おにのいす
The Oni’s Chair, 350 words

へっこきよめさま
The Farting Bride, 350 words ♬

おしょうがつさん どんどこどん
The Spirit of the New Year, 350 words ♬

きおくのおうさま
The Memory King, 400 words

にゃんこおうじ
Prince Meow, 400 words

くらげのりょかん
The Jellyfish Ryokan, 425 words ♬

ねえ、おつきさまどうしてぼくについてくるの？
Say, Why Is The Moon Following Me?, 450 words

おばけにょうぼう
Ghost Wife, 450 words ♬

ともだちやもんな、ぼくら
Because We’re Friends, 550 words ♬

さよならぼくたちのようちえん
Goodbye, Preschool, 700 words ♬

なぞなぞのみせ
Riddle Stores, 750 words

星のベイビーサーカス フルーツ島のわんぱくパイレーツ
The Baby Circus of the Stars: The Mischievous Pirates of Fruit Island, 800 words

3びきのかわいいオオカミ
The Three Little Wolves, 1,200 words

みんなあかちゃんだった
Everyone Was Once A Baby, 2,500 words

USA

California

Costa Mesa

• Kinokuniya: I’ve never been to this location but I presume it’s like the one in Seattle: that is, a spectacular source for new Japanese books, including children’s books and manga.

Los Angeles

• Kinokuniya

San Francisco

• San Francisco Public Library – Western Addition Branch: Only this branch specializes in Japanese books. It is west of SF Japan town. There are hundreds of Japanese kids books. They have all levels from kid’s picture books to juvenile books. They also have adult level books in a different section. (Thanks to Wayne for the description!)
• Kinokuniya: It’s been many years since I’ve been to this location, but like the Seattle branch, it was, and presumably still is, a huge Japanese-language bookstore with a good selection of children’s books and manga.

San Jose

• Kinokuniya

Michigan

Ann Arbor

• Ann Arbor Public Library: over a hundred Japanese children’s books, spread out over the various branches.
• The Dawn Treader: used bookstore with a shelf of Japanese books; in my experience, not very many children’s books, but there does tend to be manga.

Novi

• Mirai: medium-sized store with a good amount of new and some used books.

New York

New York City

• Kinokuniya

Oregon

Portland

• Powell’s: the most famous used bookstore in Portland, with a fairly robust Japanese childrens’ book section.
• Kinokuniya

Tennessee

Nashville

• The Nashville Public Library: has about 20 children’s books in Japanese, ranging from high level 1 to low level 3. These seem to all be at the main branch. Not much, but it is a place to start. (Thanks to e_dub_kendo for the description!)
• McKay Used Books: I can’t speak for their other 2 locations, but the Nashville location is an enormous used book store, with an entire bookcase of Asian books. Generally these are about 70% Japanese. I usually only find 1 or 2 children’s books per visit, but there’s always plenty of manga. Prices are incredibly cheap. Generally I pay less than $1 per book. Sometimes you can find manga and light novels in really decent condition being sold for .25 cents. Definitely a resource worth checking out. (Thanks again to e_dub_kendo!)

Washington State

Pierce County

• Pierce County Library: there are around 800 Japanese books of all levels spread among the various branches, though I can’t say how many of those are children’s books. The Lakewood branch and University Place branch seem to have the most.

Tacoma

• Tacoma Public Library: currently has around 125 Japanese kids’ books at the main branch.

Seattle

• Nikkei Bunko: If you are at all near Seattle and at all interested in tadoku, visit this place as soon as humanly possible. They have hundreds of children’s books and manga to lend, and the volunteers Nara and Bruce are extremely welcoming.
• Seattle Public Library: over 700 children’s books, spread over the various branches; most at the Central Library or International District/Chinatown Branch.
• Kinokuniya: a huge Japanese-language bookstore with a large selection of manga and many kids’ books.

Pros:

• Because they’re divided by grade, you already have a good idea of how difficult they’ll be.
• I’ve read two of these kinds of books; they were both level 3 by my system, but both of them had more content than the average level 3 book, which usually has around 1,000 – 3,000 words: “New Stories That Linger In The Heart for First Graders” had around 5,300 words and “Heartwarming Stories for Second Graders” had around 4,700. So at least at the lower grades, one of these books will most likely last you longer than an average book.
• They seem to be fairly widely available (I haven’t checked every book on my list, but Kinokuniya had both of the books that I’ve read), not too expensive new, and if you can get them used many of them are extremely cheap.
• They’re divided into different stories by different authors, so one book gives you not just varying subjects to amuse yourself with, but also examples of how short stories in Japanese are written and what different writing styles are like.
• There are fewer pictures; depending on your reading level this might be a good thing or a bad one, but it does mean that there’s more room for words.
• There are different kinds of collections: some based on literature, some about science, and even some about ghosts.
• If you try one book and particularly enjoy it, finding the next book is as simple as moving up a grade or seeing if there are more books at the same grade in a related series; you might also be able to look up other books by an author who particularly caught your attention.

Cons:

• Of the two books I’ve read, not all of the stories were extremely interesting in and of themselves, and I’d go so far as to call “Heartwarming Stories for Second Graders” boring (and my threshold for being amused is generally pretty low). I am sure that some series are better than others, but I think that these are probably the kinds of things you read to add to your general ability and not so much for their own sake. The books generally felt, to me, like extensions of what kids might read in school, so they had that vibe of “what adults think would be beneficial for proper child development.” That can be a good point if you’re interested in Japanese educational culture or want to try to mimic that experience. (If you want fart jokes instead, my Zorori series review is right this way.)
• Just because these are for kids doesn’t necessarily mean they’re easy: I would recommend that someone new to extensive reading get some experience reading shorter stories or graded readers before tackling even the ones for first graders, because with less pictures and more text, even the book for first-graders I read was on the harder end of level 3.
• e_dub_kendo points out that that they have the potential to get pretty repetitive, especially if you pick up some of the more specialized ones like the collections of fairy tales.

I’m going to list as many of these series as I can find, but I’ve only read two of these books: if you read any of them, feel free to send me a short review with what kinds of stories it had, what level it was, the approximate number of words and your rating. Also, if you find any other series like these, let me know and I’ll add them.

I’ve added Amazon links because it’s easier to collect them all in one place this way, I like to be able to see the covers and I certainly wouldn’t complain if someone used the links to order through (as an affiliate I get a percentage), but check around for the most cost-efficient way to buy before you actually order any of these, because the odds are good it’s not Amazon. Click here for suggestions on where to find these books.

新心にのこる◯年生の読みもの
New Stories That Linger In The Heart For 1st-6th Graders
The book for first graders had ten short stories, and the subjects were varied: there were standard pieces of short fiction, a non-fiction piece, some folk tales and a story by Nankichi Niimi, a famous children’s author. (This one: 一年生たちとひよめ. It was unabridged, but had fewer kanji.) Had a soft cover, so would cost less to ship if shipping costs are calculated by weight. ★★★★☆
Click here for other suggestions on where to find these books.

なぜ?どうして?科学のお話　◯年生
Why? How? Scientific Stories for 1st-6th Graders
I haven’t read any of these, but Kanjiguy highly recommended the one for first-graders. Each story was about 2-3 pages long, so with 183 pages that’s quite a few stories!
Click here for other suggestions on where to find these books.

１０分で読めるお話　◯年生
Stories You Can Read In 10 Minutes for 1st-6th Graders
Fembassist has been reading these, and says they’re mostly short stories from Japan.
Click here for other suggestions on where to find these books.

１０分で読める名作　◯年生
Classics You Can Read In 10 Minutes for 1st-6th Graders
Click here for other suggestions on where to find these books.

１０分で読める物語　◯年生
Tales You Can Read In 10 Minutes for 1st-6th Graders
Click here for other suggestions on where to find these books.

１０分で読める伝記　◯年生
Biographies You Can Read In 10 Minutes for 1st-6th Graders
This series isn’t completely available yet, but I assume they’ll be going to 6th grade eventually!
Click here for other suggestions on where to find these books.

なぜ？どうして？　みぢかなぎもん◯年生
Why? How? Answers to Everyday Questions for 1st-6th graders
Another new series.
Click here for other suggestions on where to find these books.

なぜ？どうして？科学なぞとき物語　◯年生
Why? How? Stories about Solving Mysteries of Science for 1st-6th Graders
I guess someone has found that there’s really a market for books like these.
Click here for other suggestions on where to find these books.

心があったかくなる話 ◯年生
Heartwarming Stories for 1st-4th Graders
The 2nd grade book of this series was level 3, 158 pages and 4,700 words (est.), and contained 14 stories. The stories were all slice-of-life short fiction, and, to me, weren’t as interesting or varied as the ones in 新心にのこる１年生の読みもの; it is pretty much what you would expect from a book with this kind of title. Hard cover. ★★☆☆☆
Click here for other suggestions on where to find these books.

ほんとうに心があったかくなる話　◯年生
Truly Heartwarming Stories for 1st-4th Graders
Click here for other suggestions on where to find these books.

心にしみるお母さんの話　◯年生
Mother’s Heart-Piercing Stories for 1st-4th Graders
Click here for other suggestions on where to find these books.

読書の時間に読む本　小学◯年生
A Book To Read During Reading Time for 1st-6th Graders
Click here for other suggestions on where to find these books.

読書の時間に読む本〈２〉小学◯年生
A Book To Read During Reading Time (2) for 1st-6th Graders
Click here for other suggestions on where to find these books.

読んでおきたい　◯年生の読みもの
Must-Read Stories For 1st-6th Graders
Click here for other suggestions on where to find these books.

齋藤孝のイッキによめる！名作選　小学◯年生
Takashi Saitō’s Selection Of Classics To Read At One Go for 1st-6th Graders
Click here for other suggestions on where to find these books.

齋藤孝のイッキによめる！音読名作選　小学◯年生
Takashi Saitō’s Selection Of Classics To Read Aloud At One Go for 1st-3rd Graders
Click here for other suggestions on where to find these books.

米村でんじろうのイッキによめる！ おもしろ科学　小学◯年生
Yonemura Denjirō’s Interesting Science To Read At One Go for 1st-3rd Graders
Click here for other suggestions on where to find these books.

日本のむかし話　◯年生
Tales of Old Japan for 1st-3rd Graders
Click here for other suggestions on where to find these books.

◯年生の読みものー理科や算数が好きになる
Reading Material for 1st-6th Graders: Start To Enjoy Science And Math
Click here for other suggestions on where to find these books.

おばけ・ゆうれい話　◯年生
Ghost and Spirit Stories for 1st-3rd Graders
Click here for other suggestions on where to find these books.

(There are also older versions of these that are very cheap used: おばけ・ゆうれい話〈1年生, おばけ・ゆうれい話〈2年生〉, おばけ・ゆうれい話〈3年生〉)

科学なぜどうして　◯年生
The Whys and Hows of Science for 1st-3rd Graders
Click here for other suggestions on where to find these books.

世界のわらい話　◯年生
Funny Stories from Around The World for 1st-3rd Graders
Click here for other suggestions on where to find these books.

(There are also older versions of these, although the used price for them isn’t always better than the used price for the newer ones. 世界のわらい話〈1年生〉, 世界のわらい話〈2年生〉, 世界のわらい話〈3年生〉)

グリム童話　◯年生
Grimm Fairy Tales for 1st-3rd Graders
Click here for other suggestions on where to find these books.

アンデルセン童話　◯年生
Andersen Fairy Tales for 1st-3rd Graders
Click here for other suggestions on where to find these books.

ことわざ物語　◯年生
Proverb Tales for 1st-3rd Graders
Click here for other suggestions on where to find these books.

(There is an older version of this series as well, which may be cheaper used: ことわざものがたり 一年生 , ことわざものがたり 二年生 , ことわざ物語 三年生 )

イソップ童話　◯年生
Aesop’s Fables for 1st-3rd Graders
Click here for other suggestions on where to find these books.

世界の名作童話　◯年生
Famous Children’s Stories from Around The World for 1st-3rd Graders
Click here for other suggestions on where to find these books.

親も子も読む名作　◯年生の読みもの
Masterpieces for Parents and Children: Reading Material for 1st-6th Graders
Click here for other suggestions on where to find these books.