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If an action rather than a stative verb like belong to is required, I'd suggest:

• (Person A) goes to (club)

This does not denote an action performed within the club premises but indicates that the person takes part in activities carried out there.

Attends.

There isn't a really good answer for this, since there's not a strong word for what exactly you do at a club. And frankly, that's understandable, because clubs have so very any different activities. For instance if it's a flying club, you could have

Joe flies at the club

However, what's usually the case with clubs, is that you show up and then do whatever the club does. So this is a viable catch-all:

Joe attends the club

Participates

Joe participates in the Drama Club and the Chess Club.

(I suppose it's possible to be technically a member of a club but never actually participate in any of its activities, but I'm going to overlook that since I can't think of a word that would cover that situation :))

I think the best verb for this, which has two senses that apply, is patronize:

1 : to act as patron of : provide aid or support for

// The government patronized several local artists.
3 : to be a frequent or regular customer or client of a restaurant

// much patronized by celebrities

In terms of a club, if you are a member, then you have paid your dues in order to support it (following the first listed sense of patronize) and be able to have access to it as a customer or client (the other listed sense of patronize).

In your sentence, the preposition would be dropped:

(Person A) patronizes (club).

Perhaps Groucho Marx said it best? (Emphasis mine)

I don't care to belong to any club that will have me as a member

Another I've heard is "active member," "active with," or simply "active"

She's active with the Latin club

She's active in the Latin club

She's active Latin club (slangish)
Active duty assumes military.

Also, a bit more humorous with implied advocacy is card-carrying member. This is beyond membership; it’s for those committed to the point where they carry a current (dues paid) membership card in their wallet/pocketbook:

She's a card-carrying member of the National Rifle Association

..she religiously attends and is an active participant in all events, including the regularly held third-Tuesday-night-of-the-month business meeting from 7-10pm. People like this will often happily produce and showcase said membership card upon request, and proselytize upon the virtues of their organization. :^)