There is no official rule, but generally speaking, roles on Broadway are not given to actors who are under the age of 4. Historically, children much younger than this have taken part in big shows, however standards have changed, and this practice is now generally unaccepted. For parents who want their children to become actors, then attempting to get them a role on Broadway while they are small can be an excellent way to kick-start their career. After all, some of the most famous actors of today were once child actors.
The job roles for children on Broadway are limited. This is mainly because children are only really used in junior productions where the cast is entirely made up of those under the age of 18. If the adult acting industry is competitive, the child acting industry is even more so. So, you shouldn’t expect it to be easy if you want your child to get a role on Broadway. If you want your child to achieve one of these roles then it is best to get an agent to help you get auditions.