Did you already write it? Why not send it to as much publishers as you can find, for example this wikipedia list: http://en.wikipedia.org/wiki/List_of_publishers_of_children's_books. I’m not sure if that’s safe and they won’t steal your work though, someone else might have some insider knowledge of that.
A speaker at this month’s writers’ club meeting remarked that the children’s book market is just about the hardest to break into. There are a lot of ways to do homework in this field. Starting with the library and Writer’s Market is one good idea. Online research is another. Don’t stop with the first answer you come to.
The speaker I heard said that he finally self-published and went around reading his book to kids in schools. He had a lot of fun with it but never made any money.
I do remember reading that publishers don’t want authors to send illustrations with their stories. The only exception is when a sketch is absolutely necessary to get the idea across. They want to assign the art work themselves, and the author won’t have any say in it.
@Jeruba , that’s an interesting observation. Most of the children’s books we have, the better ones, anyway, are illustrated by the author, e.g., Dr. Seuss, Five Little Monkeys (Eileen Christelow), Richard Scarry’s Busy Day books, etc. Most of these books have been around for a while, but it would suggest that being an illustrator as well as a writer is actually an advantage.