Is there a legitimate reason to not want to pay via amazon gift card?
If they use Amazon account that they share with their family, there will be a record of the gift card in orders.
If they open a new Amazon account for a purpose of sending the gift card, the new account might get automatically blocked.
If they are concerned about their privacy, tell them they can buy a physical Amazon gift card with cash and send you the photo of the code, or just the code.
Remember to add it to your account straight away, so that they won't be able to redeem the code themselves (it applies to the e-gift card as well, just redeem it straight away).
If they just hate Bezos, tell them they can buy another gift card. You can ask for a gift card to a shop where you buy your groceries regularily.