Okay... here is MY personal suggestion. This is the Publix on the corner of 192 and South International Drive, near the Gaylord Palms. It is the closest grocery store to the Boardwalk. If you hit every light, it'll only take you 15 minutes.
When you exit the Boardwalk, take a RIGHT.
At the traffic light, make a LEFT onto Buena Vista Drive. Get into the right most lane.
At the next light (you'll see Reedy Creek on the left), make a RIGHT onto Victory Way. You'll pass Pop Century.
At the next light (You'll see Wide World of Sports), make a LEFT onto Osceola Parkway. Stay in the right most lane.
At the second traffic light (intersection of Gaylord Palms) take a RIGHT onto South International Drive.
Stay in the left most lane. As soon as you go under the overpass, there will be a Publix on your left.
To get back:
Take a RIGHT out of Publix onto S. Int'l Drive, and get in the leftmost lane.
At the traffic light (Gaylord Palms) make a LEFT onto Osceola Parkway.
Once you pass through the archways, get into the right most lane.
At the traffic light (Wide World of Sports) make a RIGHT.
When you get to the Reedy Creek station, make a LEFT onto Buena Vista Drive
At the next traffic light, make a RIGHT onto Epcot Resorts Blvd.
Make a LEFT into Disney's Boardwalk.