It is a little bit of a pain to refill the mugs at the Bakery, but now there is a stand on the Boardwalk called "Boardwalk Joe's" at which you can fill your mugs yourself with coffee. The trouble was that it isn't open as early as the Bakery, so most times we were heaed to the parks before we could use the self-service cart.
The mugs can be filled poolside and at the indoor food/beverage area off of the lobby - but I'm not sure what kind of beverages are there as we have never used either place.