How would you design a table for approval system between User and Sports Club
Add a self-reference to User. Call this reference "approved by". I'd suggest implementing this as a separate table but if you're absolutely sure you'll only ever need three approvers it can be colums in the user table.

As a current member approves an application count the number of approvals, compare it to the number of required approvals, and the number of current members and proceed accordingly. Remember to account for the case where the current applicant is the first to apply to a club.

I think it's an error to make a user exclusive to one club. Wouldn't it be good to know that the same person plays polo, frisbee, netball and Jai-Alai? It would be better to have an intersection table called Membership.