There are business centers at the Dolphin, YC/BC, Coronado Springs and Contempory. None of htem are free, but you can use them to check email. The Disney hotels have normal business hours. The Dolphin has a 24/7 deal. You just swipe a credit card in thier machine and go.
Also, if you have a car there is a Kinkos in Lake Buena Vista I have used and probably a public liabrary someplace that's free.