When I go shopping, I use one of these:
https://www.comptoirdelavannerie.fr/produit/couffin-en-palmier-poignees-en-cuir/
It's made from natural fibres and the last one lasted 15 years. When it starts to break up, I use it as a plant container and it eventually ends up in the compost. The current one is already 10 years old!
In the basket, I carry a couple of those:
https://www.filt1860.fr/fr/12-filet-a-provisions-petite-poignee.html
Also, some home made pull string bags for loose ingredients, like rice, beans flour, etc.
Most food retailers in France leave empty cardboard boxes by the tills for the customer's use, very handy for bulkier items. Then of course they are very useful for storage, lighting the fire. mulch or compost
