As stated in this answer, it depends on the airline. In your case however it seems you won’t be asked for a credit card. However, I have a general piece of advice for you, based on my personal experience. When in doubt, you should carry a scanned copy of the credit card, and a declaration, signed by the card owner, which states that you are allowed to use it. Make sure you don’t loose the credit card scan, of course.
While a few years old, this article on how some airlines require credit cards at check in states that American Airlines does not have this policy of requesting the card at check in.
And given you can now even print your boarding pass at home, there certainly doesn’t appear to be a requirement for this.
Short answer: no.
Even paying with a credit card at all is optional as American Airlines accepts PayPal.
Your mother may be asked to present a credit card if she is presented with the option of an upgrade and wants to take advantage of it and she may be asked for a credit card if she wants to buy snacks on the flight but she does not need to present the credit card that was used to purchase the tickets.
Credit:stackoverflow.com‘