From Animal Kingdom Lodge:
Drive to the exit of the lodge. Make a right at the light (Sherberth Rd). Follow that to the next light (US-192). Make a right. Go about 1.5 miles. There will be a Super Target in the plaza on the left, or a Publix in the plaza on the right.
(For a 24 hour Super
Walmart, it's about 5 miles on 192 until it ends, then briefly North on US-27, and the Walmart is on the right)
Edit: These are the closed to AKL. There are others in the area, closer to other parts of Disney property. AKL is pretty much the far Southwest corner of Disney Property. A lot of people don't realize that WDW covers an amount of space roughly equal to the city of San Francisco