You have a couple of options.
1.) There are car rental desks at the Downtown Disney Resorts. There's a
Budget at the Doubletree, a Dollar at the Holiday Inn (there was one there when it was the Courtyard, I don't know if it's still there) and an
Avis at the Hilton. You could rent a car for a couple of days. You could then in that time go to a grocery store if you don't have a grocery stop with a towncar service.
2.) A taxi - not sure how many of you there are, but I think cab fare would run you about $20-$25.
3.) Mears runs shuttles. I believe it's $12 roundtrip for adults, a little less for kids.
4.) Lynx is the cheapest way, at $1.50 per person I think, but the routes are very convoluded, and involve a couple of transfers.
I personally would go with #1, but if that's not an option, then definitely a taxi. You have to call Mears to arrange for the transportation 2 hours before you're ready to leave. The on-demand capability of the cab is worth the extra money as far as I'm concerned.