I just got the DDE while at WDW, so here's what I had to do....
I had a MYW ticket that I wanted to upgrade to an AP, adding the DDE. I had to do the ticket upgrade first thing in the morning (I arrived very late at night), then called at 9am about the DDE to sign up and give them credit card info to pay for it, then picked up a temporary card at EPCOT's Guest Relations window after 5:30pm that day. (This is the only Guest Relation window that can do this apparently. You want the window outside the gates, not the office labeled Guest Relations that is inside the park.)
This seems to be the quickest way to get the DDE at the parks - like if you are getting your AP there as well. You do lose the discount on your first couple of meals since you don't have the card yet, but that's not a huge problem. If you already have your AP, you can call any time, though it's active from date of purchase, not first-use.