There are several grocery stores not far away from where you'll be staying.
Go from your resort to Downtown Disney and turn onto Hotel Plaza Blvd (this is the road that has several non-Disney hotels on it). Take that to the end (it ends at SR 435).
Now you have a choice, left, right or straight.
If you go straight across the intersection into the parking lot, you'll see Goodings "grocery store" on your left after the McDonalds - I'd suggest staying away from this store, as it is VERY expensive, sometimes more expensive than getting things from your resort.
If you go left, drive down less than 1/2 a mile, and you'll see a Winn Dixie on your right (it's across from a Walgreens, I believe).
If you turn right, then go under I-4, and take a left at the first stoplight onto Vineland Avenue. Take Vineland past the restaurants and hotels, the church, and the outlet mall on your right. After the outlet mall is a Publix grocery store.
The Winn-Dixie and Publix above are both less than 10 mins from Downtown Disney (more like 5 minutes); we've used both, and I think I prefer the Publix, I think their produce is a bit better, but that may change from day to day.
There is another Publix grocery store MUCH farther down on SR 435 (right turn off of Hotel Plaza Blvd, then go a LONG way down SR 435) - it's at the intersection of Rt. 435 and 192.
Hope this helps!!