Not totally helpful (but related). I live in a city where any track that you'd want to run on is locked up. But you kind of learn which ones are the easiest to get into and whether people will chase you off. I've been asked to leave many times. But never anything more than that (just be nice).
I was up visiting my in-laws for the holidays and there's a public college nearby. The track there was completely unlocked so I went there with the kids (and dog) every day (even parked illegally without a parking pass). On day 2 we were kicking a soccer ball on the infield with our dog nearby on a leash tied to the goalpost. 2 police officers come walking into the stadium. They look at us but sort of just keep walking. Then about when we think it's clear they break stride and head right toward us. My wife mumbles something to me along the lines of we're about to get kicked out. When they get close enough to us to talk one of them asks if they can pet our dog. So we chat a bit and then they go on their way. About 5 minutes later one of them comes back to let us know that he unlocked a restroom for us in case one of us needs to use it. If I was at home and that happened 100% we would have been asked to leave. But it was a very welcomed surprise and just goes to show that not all communities are the same.
Regarding your original question. If you want to ask I would start with whoever coaches the distance runners on the track team. They are most likely to be sympathetic to your situation. I would say there's a very high chance they will tell you no (or just not respond).
Personally, I would just use it and try to be as discrete as possible. Try to make friends with whoever you see out there at that time and hope that they don't report you. For the high schools we sneak onto in our area I always get my kids shirts from the school and then have them wear those. Unless it's someone in the know most people just assume we're legit because of that.
Most coaches and athletes will be welcoming of anyone interested in their sport and will help you out as long as you're not interfering with their practice. It's more like the AD or principal that you need to be concerned with. Maybe I'm lucky because I'm on good terms with the school administration after I developed a project with free essay samples (read more samples here) for students to take apart assignments by example.But I am sure that in the case of adequate people - you can find common ground and avoid any conflict situation.