Hi, Peter!
I am stumped on this one! I just walked through buying tickets to see what you might be doing wrong, but I agree with you - I cannot see a place to use your Disney Gift Card (
which you can absolutely use to purchase your theme park tickets.) I have often used a Disney Gift Card to pay for my Disney vacations on the site, so I do not know why buying "tickets only" would be any different. That said, there is no space for that gift card number!
I do not know if there is some weird rule about buying tickets online with a gift card, but to simplify, call a Disney reservationist at (407)W-DISNEY and just purchase them over the phone. They will be able to use your Gift Card number as payment. I am sorry I do not have a better answer, but I have tried to do this several ways, and the only options I am seeing are the Disney (Visa) Rewards card or Credit Card.
You will be asked your method of delivery, so be ready to let them know if you want to pick-up the tickets at Will Call (free) or have them mailed to you (small fee.)
However it works out for you, I hope you get your theme park tickets without a glitch and you can get moving on planning the rest of your magical Walt Disney World vacation!