It depends on the ticket. AA does not require the credit card used to purchase the ticket to be produced. But some other airlines do. For example, BA requires that prior to flying the first leg of the ticket, the credit card used to pay for the ticket be produced. So if you buy a ticket on aa.com and the first sector is BA, then the credit card might be needed before travel is allowed (I say might because I suspect BA has some leniency with this policy, but have never tested it). Many other airlines also have this requirement, not just BA.