I'm happy to announce that I've found a farm stand that suits my needs. Urban Oaks is an organic farm right in New Britain, just a few minutes from my local Stop and Shop. They have greenhouses and sell fresh organic fruits and vegetables grown on site year-round. They have farm stand hours on Friday, which is my grocery shopping day. That means that I can go to Urban Oaks, pick up whatever fruits and vegetables I need, and then get the rest at S&S. I also love that they carry frozen grass-finished beef (from a farm in Vermont), as well as some other local products like baked goods, honey, orange juice, etc.
(I had visited a couple of farmers' markets in West Hartford, and although I got some great stuff there, it felt like more of a novelty market than anything practical. Plus, there was no grass-finished beef to be found, not even inside Whole Foods.)