No, legally they can not hire a 15 year old due to labor laws in the US. You may be able to do volunteer work at an animal shelter for no pay, but they can not hire you on as staff and pay you.
You'd have to call animal shelters near you. Ask them the age limit. Just put in google "animal shelter" and then where you live. your city, and state.
No they don't hire that young but i do know they accept you to volunteer?