I know several active San Antonio area runners, including one accomplished and actively competing triathlete. Despite my personal experience to the contrary, though, I don't think San Antonio has a very active running culture. The heat and humidity probably have something to do with it, as well as the city's general predisposition to the notion that sports = football and not much else.
Although school's back in session, I can't imagine some of the school tracks wouldn't be accessible when not in use by students. For example, Justin and I went to Reagan High School (NE side) a month or so ago to launch battery-powered RC planes. We used the football field after a father and son wrapped up their two-man soccer match. The track and field were accessible (not padlocked).
Your best bet is probably to hook up with a local running club and see where they do their runs. They'll have sorted out the various access issues and probably have other recommendations for running sites near your home, as well.