I don't think it's available yet...
I think that e-readers for kids just aren't made yet, but if you are willing to give it a try, you could get a used Kindle on Amazon for a reasonable price. By getting the used product you may be giving up a guarantee that it will work for very long, but it can at least be painless to lose.
I don't think it will be long before someone comes up with something that is appropriate for children. Keep your eyes open.
Oh... and don't forget...many classic books are available for free. There is certain to be a large selection of books for you to choose from for kids and adults. No need to even concern yourself with buying books, unless they are books you specifically want to buy.
Note: This post was edited by its original author to combine 2 posts into one on 06/21/2013 at 5:23 PM PT