It is possible. I'm sure there are lots of suburban and rural locations where it would be more difficult. But here in Atlanta, it's pretty simple: Put some sort of racks on your bike, whatever works for you. Ride to the store. Use a basket rather than a shopping cart, at least until you figure out how much you can carry safely. Secure bags to bicycle. Ride home.
Apparently, a group in Portland, Oregon offers a class in bicycle grocery shopping. Do we need one here?
A one-bag shopping trip