Vulcanized Latex RubberExactly modern grocery bags, no
Exactly the same, no. Modern grocery bags are made from very precisely refined polymers which require thermometers to get to the exact properties you would be looking for. Thermometers are a post medieval tech; so, even with a perfect understanding of chemistry, you can not refine elements to exact temperatures without one. If you can't control temperatures exactly, then you can't manipulate your base materials to get pure and precise chemical reactions to get plastics with exactly the properties you want. Because grocery bags are so thin, even minor impurities would cause them to fall apart.
Something similar to modern grocery bags, yes.
Latex rubber was invented around 1600BCE by the Aztecs and is forgiving enough to work with without a thermometer, butand various kinds plants that could be refined for the stuff exist all over the world. The issue with natural latex is that it was no stronger than an eraser until people figured out how to vulcanize it by heating it with sulfur. Natural sources of relatively pure sulfur were known to many ancient civilizations such as the Greeks and Chinese as early as ~600-500BCE, and the Chinese learned to refine it around 300BCE.
With access to relatively common resources and a little bit of experimentation, it would have been as simple as boiling the resin of the right kind of plant with the right kind of rock and you'd have vulcanized rubber. Your bag's properties would more closely resemble a ballon than a modern grocery bag, but it would be a biodegradable polymer based sac that could serve the same function as a grocery bag.
That said, inthese ballon bags would be possible, but not the most probably application of the technology. In the pre-industrialized world, almost nothing was disposable. Everything was so much effort to make you would not just throw it away when you are done unless it was actually ruined by its intended purpose (like wine skins); so, your people would probably use such as bag more as a satchel or tote bag than something they hand out for free at the marketplace. So, a more realistic application of flexible plastics in the ancient-to-medieval world would be to use whatever plastic or rubber you have and use it to waterproof a linen or leather satchel so that the base material can give it structure and the polymer can make it waterproof. This way, even if it is technically biodegradable, it will still give you a simi-permanent useful item for your effort of making it.