Chris,
No, Disney does not run shuttles to the grocery stores. However, while you may not easily find a way to get to the grocery store, there are a few reasonable options to get the groceries to come to you!
A nearby grocery store, Goodings, will deliver your grocery order to your hotel. I have not used it myself, but if you visit their website at
www.goodings.com you will see how to order. There is a minimum order requirement, as well as a fairly significant delivery charge. It appears that if you are ordering a lot, it might be worth it.
Another choice is
www.gardengrocer.com. Again, I have not used this myself, but it appears there that you have a lower delivery charge, and can get that waved with very large orders.
If you are set on going yourself, I would recommend a cab. It will save you time and maybe money in the long run. I would plan on about $20 each way, just to be safe, depending on what resort you are coming from.
I hope that this helps?
Have a magical vacation!