Yeah, I sent an email to their "Contact Us" and this is the response I got:
Thank you for contacting us with your inquiry. Unfortunately, the Disney Gift Card transfer functionality on DisneyGiftCard.com is currently down for maintenance. We are hopeful it will be back in operation by early next week.
Please Continue to check the website for the return of the transfer link. We apologize for the inconvenience.
Sincerely,
Disney Gift Card Guest Service.
I guess I will just end up calling to pay our bill and read them a bunch of numbers over the phone. "Oh Joy"