Typical plastic grocery bags have been used at markets I have shopped on STJ. I imagine that you can bring along your reusable bags if you wish to pack and carry them.
You don't "need to', but it's always a good idea, especially here with almost no recycling and our overflowing landfill. (Well, not really a landfill, just a dump)
As Hugo points out, it is (always) a good idea. We still have a fabric bag that we were given on a trip to the Marketplace by First Bank in an attempt to encourage re-use. As a result, we always bring some of our own re-usable bags with us when traveling especially when it is a destination like StJ where we will be grocery shopping.
I usually bring down a couple of large canvas totes that I use for beach bags and they double as grocery bags too. Quite often you'll find a cloth grocery bag under the sink at the villa. I usually take a cooler to the store too to keep the cold stuff in.
Great idea to bring reusable bags. We have always kept the plastic bags and used them for dirty clothes, then brought them back home to recycle. I keep a stash of bags under my sink to use and I love it when I pull out a Starfish one every now and then! Next time I'll bring my reusable bags...