Writing Children's Books and Magazines: Check out Children's Writer Newsletter of Writing and Publishing Trends. http://www.childrenswriters.com/ Click on Writing Contest - There are several and they must be submitted by October 31th.
This newsletter is great source to keep up with the latest news in children's book publishing and magazines. They provide markets and guidelines and articles. Check it out. Also, don't forget to go to the Children's Book Council for a list of publishers and see who are excepting manuscripts.
You don't know if you don't try. Possibles can be yes, if not, remember that story you sent in may not have been written, if you hadn't tried. Gathering experience is an important part of growth in becoming a professional writer.