In Providence, your best bet is the Urban Greens buying co-op
. They've been getting organic, locally grown squash, apple, kale, cauliflower (to name a few) and local milk and eggs. You can order online and orders come every other week.
Supermarket-wise, Whole Foods seems to be doing a decent job produce-wise; their Waterman location in Providence has savoy cabbage from RI, among some other locally grown delights. But you'll need to go to Dave's/Stop-n-Shop/Eastside/Shaws/more
for Rhody Fresh milk and other local dairy products.
In NJ, Princeton's Whole Earth Center (360 Nassau St.) always has a list of what produce is local written on a board when you walk in.
Give a call and you might be happily surprised (or at least get a tip on where else to look). Such is the case at Merrick Farm
in Farmingdale, NJ. Susan and Juan run a CSA during the summer, and this winter they have been growing lettuces, cilantro, fennel, garlic, brussel sprouts, and more in their greenhouse. Right now they're just barely covering heating costs, but as the word get out, they're hoping to be able to expand the varieties and amount of what they grow. Local greens in the midst of a snow-covered Northeast!