Chapter 3 of our French textbook includes the following supermarché vocabulary:
le chariot: shopping cart
le sac en plastique: plastic shopping bag
This chapter focuses on Québec, not France. Still, for our young French learners, we should mention le panier, the ever-popular basket with handles that typical French folks use to pick up an armload of provisions at the bakery, the deli, etc. And do note that the word comes from saying "the thing you put your bread in"!
We should also mention that the French government has for the most part banned plastic shopping bags, reducing the number of plastic bags used annually in France by 85% since 2002.