Depends on how you define "Published." Many kids contribute short letters or essays to magazines and newspapers, both in explicit "children's sections" and outside of it. As for short stories, again, some authors are quite young when their first works are published, although they may not be published by a "professional market" in terms of pay rate and audience. Novel authors like Amelia Atwater-Rhodes and others can be quite young when their books are first published. Definitely 16 or younger, sometimes as young as 14. Bethany Huang published her first novel, The Eiffel Tower's Daughter, shortly after she turned 11.
were either/both of those vanity published?... or were their books put out by traditional paying houses?
About 30yrs ago a nine years old girl wrote stories about an onion and his vedgtable friends. It was adapted for TV and went out on, what at the time was called, childrens hour.
Mama - Atwater-Rhodes was in the unique position of having a middle-school English teacher who was also a literary agent. I believe he was able to get Random House to pick up the ms when she was still 14. Of course, there was likely some value in the PR of signing someone so young. A simple case of "who you know". -Frank
There is also Nancy Li Fan's Swordbird. She was 13 when the book was published by HarperCollin's Children Books. I think she began writing it when she was like 11 and finished when she was 12. There is also two books a prequel and a sequel I guess.
Contacts is very important in this regard. If you can make a lot of money (someone in the know can see that), they will pick you up fast.
Don't know about Huang's book, but Amelia Atwater-Rhodes was picked up by a traditional publisher of YA fantasy fiction when she was 14 -- I assume you've heard of Bantom / Doubleday, part of Random House? -- and had several decently-selling books out before she was 18. I've read about four of them, and thought they were pretty good. She's not the only under-18, of course. There's a girl, I want to say her name is Moskowitz, who published Break and another YA book before she was 19. They've sold quite well and Break won an award as a good book for teenagers.