There is a specific office that handles the DDE. You can't sign-up in person at WDW, rather you must call the DDE line during business hours. I know the days are M-F...not entirely sure of their operating hours.
It reportedly takes 4-6 weeks to get the actual card in your hands, but DDE will fax a confirmation to your resort that you can use the rest of the trip to obtain the discounts. I've seen many posts from people who have had no trouble using the email to obtain the discount. I'm sure a LOT of folks have been joining DDE since this opened-up about 2-3 months ago, so the waitstaff is probably used to seeing the email confirmations by now.
So, if you're arriving on a weekend, you might lose some discounts. But if you don't mind waiting until Monday morning to call, you should be able to get the confirmation faxed to you and be set for the rest of your trip.