If you don't mind going through the "backstage" section of the MK area and heading up via Reams, this is closest:
7880 Winter Garden Vineland Rd
Windermere, FL 34786
I believe there's no actual problem with taking that route. You just leave the parking lot and go right on World Dr., then right on Center Dr, then right on Reams, and left on 535 (Winter Garden Vineland Rd.). The Publix is on the left a little ways up the road.
You'll pass by a bunch of cast service areas and parking lots, but I've driven back there; there aren't any gates that I know of.