There are a number of Publix in the area. Here are the directions to my favorite Publix.
Exit the Boardwalk following WDW signs to the AKL. At the intersection just before entering the grounds of AKL take a left on Sherberth Rd. Follow the road until it ALMOST intersects 192 taking a right on to Black Lake Rd. (there will be a Holiday Inn on the corner). Follow Black Lake (it runs parallel to 192) until it intersects 192. Take a right (west) on 192. Follow the road until you come to Publix on your right. There is a second Publix, a bit further down on the right also.
These two seem to be the cleanest, largest and best stocked of the many in the area.
There is another new Publix near the Premium Outlet mall and Mary, Queen of the Universe Shrine.
Exit the Boardwalk and follow the WDW signs to DTD. Take a right on Hotel Plaza Blvd. Follow this road until it intersects with 535 (at the Crossroads Plaza). Take a right and immediately move to the left lane. Just beyond the I-4 overpass take a left on Vineland Rd. Follow this road (will pass in front of the outlet mall and shrine) until it "t's" taking a right at the "t". Publix will be just down the road on your left. (It is essentially behind the outlet mall-to the left).
Hope this helps!