I am not sure this is the closest place, but there is an Albertsons we always stop at off the toll roads which is about 5 minutes away from the WDW property. There is a closer grocery store just on the out skirts of Downtown Disney called Goodings in the Crossroads shopping center, but the prices are much higher.
Yes just turn onto Sherberth Road as you come out of the Lodge and then once at the 192 junction turn right and it is up the road on the right. Nice and easy to get to from AKV.