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Since you don't have a membership/member card, you don't have access to the gym.

Hi,
Do both membership and member fit in the above and mean about the same to you? Thanks.
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In my experience, "membership card" is the usual term. I'd readily understand "member card", and I'm sure I've come across it from time to time, but it's not an expression that I would use myself. A quick look at Google makes me wonder if there is an AmE/BrE difference here though, with "member card" possibly being more common in America?
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I would say that it clearly depends on the gym to which you have a member/membership card.