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