I would look for an athletic training gym that specializes in young athletes. The franchise ones in our area are Parisi, Athletic Republic and Red Line. I know of at least two independent ones here as well. My boys have done Parisi and DS2 wants to check out Red Line since some teammates are working out there. Athletic Republic is further away so we haven't done anything there. I like these gyms because they work with a kid's age, development and goals to design a program rather than a one-size-fits-all larger fitness center.
While many of the kids end up at these gyms to improve their sports performance, there are definitely kids there to gain strength and fitness alone.
Mom to Two Wild and Crazy Boys and One Sweet Baby Girl