In regards with working at a barn with other people's horses, it really is up to the BO, and your ability to work around and handle horses.
I was around your age when I started working at the barn I boarded at. My situation was a bit different than yours, however, because I did not get paid. I was working off my horse's board.
But, none of the other boarders had a problem with me handling and feeding their horses. After about a year of me working there, I even watched the barn for the BO for a long weekend. That being said, I was very lucky that I only lived about 5 minutes from the barn, and my parents didn't mind driving me when needed.
I don't know your whole situation, so my advice to you would be to start volunteering at a barn first to make sure your up for the work, and then look for a job working at a barn. If you're looking to get paid, you may have a better time getting hired at 15 years old.
Good luck! I hope everything works out for you =).