I was going to say the same thing, about it being an extension of the school.
But do your homework first, before going to the school authorities. At one point you probably filled out paperwork that included the behavior policy and consequences for children who violated these policies. If there were none, well, then, the Y and the school district are grossly negligent. But I bet there are some. Point out to the Y administration how the local site is failing to observe its own policies.
(If children are a danger to others in an optional program, then just imaging what it might be like out on the playground during the compulsory education part of the day...)
And don't go to the principal. The principal probably has no authority over the staff of the program within this context. This is run by the Y, the Y probably pays for the facility. However, the school district probably has a liaison person (assistant superintendent) coordinating these programs between the Y and the school sites. Don't waste time on a principal who sees her job as being hostess with the mostest (my experience with dozens of elementary-type principals).
While I never condone bullying behavior of any kind on my watch in my high school classroom and surrounding environment, I can't be evertwhere. There are two pieces of the picture here. Some kids get picked on. I can smell them from across campus. I call them "Bully Bait." Some do obvious things, with others it's very subtle. Get your girl in martial arts. A dignity of character shows up very quickly in these kids. The subtle change makes all the difference.
Protect your baby, but realize that this is not an isolated problem-it will could show up elsewhere.
Love on that baby-girl.