The closest will depend on what part of WDW you are in. There is a Goodings right across from the gate to DTD but I wouldn't recommend it, prices are just as high (maybe higher) than shopping at your resort and the selection is not much better.
The Wal-Mart just south and east of WDW is pretty easy to get to, and close. And HUGE! And it has a liquor store in front. Aisles are wide, prices are decent, crowds are manageable.
The closest will depend on what part of WDW you are in. There is a Goodings right across from the gate to DTD but I wouldn't recommend it, prices are just as high (maybe higher) than shopping at your resort and the selection is not much better.
Been there on both of my last trips. Very easy to get to but make sure your gps is set to avoid the toll road or carry plenty of change (the road BEHIND the store to gets back to disney, it's a exact change toll road LOL luckily I had the right amount of quarters) however there is a short and easy non toll option as well. It was maybe 15 minutes from the epcot area.
Large, clean, prices are reasonable (as in I live in midamerica where price of living is low and while more expensive then home, not unreasonably more), liquor store (i didn't buy so don't know the prices), Subway (makes a great first night dinner to grab a bag of subs and get a chips and soda in the store), they also have a large disney section so a great place to pick up a few inexpensive souvieneers. I picked up water, beer, milk, chips/snacks and souvineers and spent less than 30 bucks.
We always go to either target ar walmart. Vineland rd is called 192. The 192 W there is a really nice super target and E has the walmart. You can avoid toll roads if you just follow the signs to 192. Also there is a walmart on 27. If take 192 W it ends at 27 and you turn right and walmart is on the right hand side.
We were told by DVC owners this past Aug. that there is a Winn Dixie about a mile from the hotels at DTD. They said that if you go down the DTD hotel road to take a left at the light (if you went straight you would be at Goodings). We haven't been there but will probably try it our next big trip. We have always used the Publix just down the street from Universal since the first two nights of our trip are spent at Universal.