Student or Learner
US Gymnastics requires its members, which include coaches and gym owners, to include coaches and gym owners to undergo criminal background checks every two years.
Should 'who' be used instead?
Maybe it's a BrE thing, but I would have used who there.
I would also have used 'who'; 'Which' sounds strange to me, though I would use it if the word were 'membership' rather than 'members'.
Yes, you are both right. "Who" may be better, but I still say that it does not necessarily need to be changed.
"The members of US Gymnastics" could also include local clubs, related charities, etc.
Large organisations often have smaller organisations as members.
I think these sentences, in which the members are obviously all people, are also unremarkable:
"The members of Alcoholics Anonymous, which include people from all walks of life ..." - meaning that "the members include ..."
"For this excursion, the team has five members, which include a doctor and a nurse."