As of this March, you couldn't.
However, I was having an issue with getting my gift cards because of DVA website problems, and I was worried I wouldn't get my card in time for my trip. The CM I talked to said that, if necessary, she could arrange for me to pick my gift card up at Guest Relations when I got to WDW. I didn't have to do it in the end, but it might be an option for you.
FYI, the CM who helped me was a supervisor (I ended up with her because of the afore mentioned issues). Some of the DVA CM's seem to be particularly clueless, so you might need to go up the ladder to find someone who can help you. But, feel free to tell them that "a friend" had a supervisor say that it was possible!
