I assume you are young but that does not limit your ability to act responsible and professional.
You are basically applying for a job.
I realize your goal is to work for free riding time but look at it from the barn owner's point of view. They do not know you or your riding ability. Plus they probably have kids that ride there that might want to work for riding.
You have to at least be open to the idea of working for lessons to start with (after you do some work for free to prove you have the ability).
Curious, why are you not trying to work for riding time at the barn you already take lessons at?