We have done it both ways in the BVI, but there is something to be said for the convenience of provisioning through the charter base. We always self-provision in the USVI (Red Hook), but that's a different story.
Eating where the locals eat in the BVI is probably easiest outside Spanish Town, on the road to the Baths, but forget about it in most other places you are likely to go in your charter boat. You will see some locals at Foxy's, but that's probably not the kind of venue you had in mind as a local eatery.
Once under way, we've picked up supplementary supplies on Jost van Dyke (Great Harbour), near Soper's Hole (Tortola) and both Spanish Town and Leverick Bay (Virgin Gorda). The best place was at the Leverick Bay resort, but it wasn't cheap--nothing is in the islands. You can also hunt for a local bakery, like at Great Harbour, but some, like in The Settlement on Anegada, are not considered walking distance.
BTW, we recommend taking a slip at Virgin Gorda Yacht Harbor (Spanish town), where you can refill the water tank, stock up at the waterfront grocery/liquor store and take a taxi to the Baths (instead of swimming in from the dinghy mooring). None of this is free, but you will be pleasantly surprised at the slip fee. Alternatively, you could take a long walk to the Baths and check out the local restaurants on the way. If you want a really nice dinner, go to The Top of the Baths in time to catch the sunset. We've never been disappointed by the food or the service. It isn't for the budget-minded, but the prices are reasonable for the quality.