There are two post offices. Both are on Palm Parkway (sorta).
The first one is the new location, it replaces the one in the plaza just past the Sheraton Safari on 535. I believe it is located in the same plaza as Cici's Pizza on Palm Parkway (right side). I believe it has limited hours and services, but most likely will be able to handle whatever you need.
There is a full service Post Office further down the road, on the left side. It has more services and extended hours.
I would like to mention... if you are purchasing something at DTD to mail home... most any Disney merchandise location can ship directly from that location. And, when you choose that option, often times you do not have to pay sales tax.
If you just need to mail a letter, there is a postal drop box (designed like R2D2) right between World of Disney and Basin. Stamps are available at select WDW merchandise locations.