Ok clearly not the type of place you go to expecting good coffee, but I'm here with my family so this is what I've discovered so far:
The only coffee outlet I could see is a Starbucks in Main Street, which has huge queues. If you're really desperate...
Believe it or not there are several espresso kiosks, with the first located at the front gate. Besides the usual American drip canisters they each have a 3 group Nuevo Summonellis and a single Nuevo Summonelli grinder with semi-auto tamp.
Naturally the staff are very poorly trained, but if you're specific about getting the milk steamed to 130 degrees and only half filling the huge cup, then you might come away with a drinkable latte. I think we've got 2 out of 4 so far that were ok, given the location.
Disappointingly, the Italian pavilion didn't have anything coffee related.
A similar espresso kiosk at the front gate, but I couldn't see much else inside the park. There is a coffee/bakery kiosk in the Africa section, but it didn't look promising so I didn't try it.
You might get a good coffee at some if the hotels. We're staying in a house but we've had a couple of meals in some of the Disney hotels and I've been really pleasantly surprised at the standard of food in those hotels, and I'm as much a food snob as coffee snob. So given this, I wouldn't be surprised if there's a decent barista or two in those hotels.
In summary, slightly better than I expected, but pack the aeropress!
Ok a very late reply, but as someone who travels to the US a bit I unfortunately have to say that the Starbucks virus has spread out of control there. I usually go cold turkey unless I am in LA or NY where good (sometimes, really good) coffee can be found.
Not Disney, but good coffee can be had from Barnie's Coffee Kitchen, 118 Park Ave South, Winter Park, NNE of central Orlando. A little acidic for my taste, but flat white, espresso and long black were otherwise all quite good.
Yep Barnie's is definitely a standout, they're excellent, above average by Australian cafe standards.
So when you stumble across them in the middle of a suburban shopping strip (albeit a very high class one) you can't help but be hugely surprised.