You might try Virgin Megastore at DTD. You could check with the front desk to see if they have any facilities for charging phones or not. When I checked into CR in December, I saw the front desk loaning a charger to someone. I don't know if it was one from lost and found, or one they had there to loan out. I just heard the man say it would only take a couple hours to charge the phone, and then he'd be bringing it back. I'm sure there must be lots of business travelers who misplace their chargers.
Another option, if you can go slightly off property, is to get one of the portable chargers that takes AA batteries. I bought mine at Walgreens, to have in the event of a hurricane. They have them for several different makes of cell phones, and there's always a ready supply of AA batteries to be found. I've never looked to see if the Hess convenience stores on property carry them, or if they carry them in the Disney gift shops. Mine was made by Energizer, but there are several companies who have them for sale.