Apple recently began allowing Discover Card members the ability to load their card into Apple Pay; however, the acceptance at Walt Disney World locations is not functioning at this time. The Walt Disney World Resort is aware of this issue and is working to fix it. All other contactless credit cards are processing correctly with Apple Pay. Additionally, all other payment transactions with Discover credit cards are currently working properly at all locations around the resort.
UPDATE (November 15, 2015): All Walt Disney World Resort locations can now accept Discover credit cards via Apple Pay. The previous issue, resulting in processing problems, has been resolved.