There are a few stands ON property owned by a company not related to Disney, called Joffreys. you can recognize their stands by the fact they have real espresso machines and they use white cups with purple on them.
Here is a list of where they are located:
One stand in the Contemporary lobby
One stand at the TTC between the entrance to the ferry and monorail ramps
Stand at AK over on the backside somewhere between Africa and Asia
Stand outside Tower of Terror exit
Stands around EPCOT, one outside Ellens Universe of Energy ride and others scattered around the World Showcase. One of those is near England, another near the Mexico side when you first start around the country.
Its the only place we get mochas and lattes now. I think you will like it.
Ive never understood the fascination with Starbucks. They are everywhere and not at all unique anymore.
I have a local coffee shop in my town and the Joffrey stands are as close to them as Ive ever found when out of town, unless you have been to Caribou coffee, WOW....but they will never venture to Disney,lol..
Anyways, hope that helps you