Can I use an iTunes gift card to sign up for the developer program?

No, you can't use a gift card. You might want to try a prepaid credit card. You can get them at most places that sell gift cards.

No - or at least, I couldn't.

I tried this today (2021-12-15) and the payment details page said "Apple Store Gift Cards are not eligible with some items in your order".

There was no other items in my order, other than the Apple Developer Program renewal.

And I use the same Apple ID for ADP as I used for the gift card. So for me, I could not pay for my membership with a gift card.

I'm in Australia - I guess it's possible that it might be different in other places.

EDIT - I logged a question with Apple Developer Program Support and they said "I understand your situation where you would like to pay for the program in partial gift card balance. However, we currently do not accept other payment methods other than credit card".

So I think that's pretty definitive.

Renewals may be less strict than the initial payment before you have filled out all the banking forms, tax forms, legal agreements, etc…

I always use iTunes gift cards to pay for the developer fees. Usually manage to get them with some rebate (20% often) which makes the developer account 20% cheaper.

The long version: Your developer account is connected to an AppleID, and the developer fees are charged to the iTunes account for your AppleID. If you added a giftcard or giftcards to your iTunes account, the money is taken out of your account. If you have a credit or debit card connected to your account, it is taken off that credit card. Just like any app purchases or music / video / book purchases.