My daughter is a couple years younger than you and has been helping out on a limited basis at the day camp at her barn. She doesn’t get anything in exchange. In the past, there’s been barn politics brouhaha where they wanted all the kids except those from a certain family to pay regular camp rate to help. I’m not able to afford that as a single parent. The horse budget has to go to cover leases and training. My daughter regards helping invaluable experience because teaching others helps her improve her own technique. Plus, she is just generous and enjoys helping others. Still, it is not an unusual arrangement for junior counselors and CITs to pay tuition and I believe you’d be hard pressed to find anywhere that won’t want you to pay them . A ride time exchange would be almost unheard of because most places have barely enough horses to pull off a camp and they aren’t going to want their already exhausted horses getting ridden yet another time that day.
I agree with other posters that you need to have a parent help you in the search. There are too many adults out there that will put you in a bad situation either willfully or through their own ignorance. My kids have the advantage of me being a lifelong rider myself. I’m far from a professional rider but I have worked as a barn manager in a professional capacity. I know enough to see stupidity coming. One recent example: an older adult amateur at my daughter’s barn has a horse she paid $500 for. The horse was something that could have been finished by an amateur when she bought it. Unfortunately, she’s a very nervous rider and ground handler and enough incidents occurred that the horse is too dangerous for most amateurs. She wanted me to ride it. I declined. Then she got the brilliant idea that maybe my daughter could ride it. Absolutely not. She ruined her horse by not listening to folks telling her to get help from the get go. She gets to pay a pro if she wants it fixed now. The thing that made me upset was that I could easily see her trying to enlist the help of a child like you whose parents aren’t horse savvy enough to understand that the horse is genuinely dangerous and something terrible happening.