Which is correct? Members must use their own cards, or, members must use their own card?
The correct version is:
Members must use their own cards.
You are talking about a collective group of people, which, here, is also plural. So, the noun (as the members don't share ownership of a single card) is plural too.
In order for the singular card to be correct, you would need to change the sentence to use a singular subject:
Each member must use their own card.
Note: I am using their in the singular third-person. For clarity, of if you don't agree with this use, it can be rephrased:
Each member must use her or his own card.
Or, with additional rephrasing:
As a member, you must use your own card.
Note that the use of own increases the awkwardness of any construction here. If you remove own, the grammatically correct subject-verb version sounds better:
Members must use their cards.
However, if the point is to say "don't use another member's card," then using the singular "member" with own (and card) is the better (grammatically correct) solution.
Apparently, I have to provide more evidence for my assertion.
Here is what Public Works and Government Services Canada (the leading source for grammar in the Canadian government) currently says
When to use a plural verb
When the members of a collective noun are performing an action as
individuals, use a plural verb. In this case, all or some members of
the group are doing something independently of the other members; the
group is not acting together as a unit.
The orchestra are tuning their instruments.
The cast have been practising their lines.
The flock were running off in every direction.
The staff disagree on the proposal.
In many cases, it may sound more natural to make the subject plural in
form by adding a word like members:
The members of the orchestra are tuning their instruments.
The cast members have been practising their lines.
The staff members disagree on the proposal.
Note that wherever individual ownership of an individual item is used, the item still takes the plural form—because the subject takes the plural form.
In a blog post by Geraldine Woods, the author of English Grammar for Dummies:
Collective nouns (committee, team, squad, army, class, and the like) refer to groups. How do you choose a pronoun to refer to that committee, squad, or team? When the group is acting as a unit—doing the same thing at the same time—the noun is singular and the pronouns that refer to it are also singular. In this situation, the collective noun is paired with a singular verb also (if the collective noun is a subject).
Right about now you’re probably wondering what happens when the group isn’t acting as one unit. Simple. Just break the group down into its component parts and go for plural verbs and pronouns, as in this sentence:
Some members of the squad are eating pizza while others are oiling their rocket launchers.
Now you have a plural subject (members) partnering a plural verb (are) and a plural pronoun (their).
And, I will add, a plural object (launchers).
If the question is "What is grammatically correct?" then the answer is what I have already given. If you don't like the grammatically correct answer, or you feel that it communicates some kind of confusion, then ignore it—but do so with the knowledge that you are doing something technically ungrammatical.