The closest Publix is a fairly new one -
Frpm the OKW area, take Buena Vista to Victory Way (I think), just past the CBR, and turn left, as if you were headed to WWoS. When you get to Osceola Parkway, turn left instead of heading into WWoS, and you'l soon leave WDW property.
When you get to International Drive, turn right. Just before you get to 192, there will be a Publix on your left.
Caution - once you hit Osceola Parkway, there's a lot of construction going on, but you *can* get to Publix.