As my daughter is 2 years old, my wife whose Japanese, has bought a couple series of books that I highly recommend. The first is "Nontan", which is a very, very intro-level series of books about a small white cat named Nontan and his friends. I've seen them sold in Japanese Bookstores on the West Coast among other places. They're quite popular among Japanese too and have been for generations.
For a little more advanced reading, check out the "Kogumachan" series which features a small little girl bear named Kogumachan. I like these books as they are a bit more challenging to read, but really, really cute stories. The artwork is priceless I think. This too is easy to find outside of Japan alongside Nontan, and is also well-known among Japanese.
Best of luck.