Most do, but not all. The Disney owned and operated places do, for sure (Homecoming, Jock Lindsey's Hanger Bar, for example). The non-Disney places may not. It's up to the restaurant. (Same thing goes for the stores, btw). Whether a place takes them or not can change from time to time as well.
To accept giftcards, or the DDP, or charge to your MB, the establishment has to be connected to Disney's computer systems - and Disney charges a fee for each of those connections (go figure!).