I believe the majority of food stands at Disney World use a Nescafe branded reconstituted liquid coffee product so that there are only fountain-beverage-style machines behind the counter -- less distraction and training for the staff.

It looks like questing for real coffee would go badly in the Magic Kingdom but decently at Epcot or Downtown Disney or in the resort hotels. This list purports to run down the places that make coffee from beans, but gives few details.

Best of luck!