It could be something like a comic book club, a book club, a friendship type of club, or what else.

7372213
I have a club base around sport

Aside from anything involving modern technology (unless AU), I'd say any kind of club that you find in a school environment with equestrian adaptations like a gem club for dragons or if you want to stretch things, a stair club for Silverstream hippogriffs.

I'd imagine they have a chess club, because all schools seem to get one sooner or later. If we're naming specific sports clubs, a fencing club is another that comes to mind.

The short, "Mystery Voice," also lists a few specific examples of canon clubs at the school:

I figure there has to be some sort of abstract "flying" club thats split between racing, stunts, and airborn sports.