We need to shop at whole foods for special dietary items, how close is it to the boardwalk villas.
Because of where BLT is located within WDW property I don't think anything is "close".
We still go out to the Publix on Vineland (used to go to Winn Dixie but learned better). I think there is a Publix towards Windermere if you wanted to venture out that way and another out past AK (off 192 ?) but I would expect it to be just as far if not farther.
The Publix near Windermere could be quicker to get to depending on traffic.
Publix is def the closest. Ask bell services, they'll give you a map and directions. You turn right out of BLT and see a lot of backstage area. There's also a yummy Italian restaurant near the Publix. It's becoming our first day tradition. I like to get settled in the first day. Then it's all about Disney the rest of our stay.