Why don’t you bring plastic bags from home? They are very much reusable, you don’t have to throw them out. They are also quite easy to fold into small shapes and keep on you, or your car, or whatever. I have plastic bags which have endured for literal years. I also decided early on that if I forget to bring a bag, I either do without or have to go back to get one. You start remembering really fast after a few times of forcing yourself to go back.
Another thing you can do is just take a cardboard box from some product in the store. This may depend on country, but where I live the shops leave products on their transport boxes on the shelves. Walking around the store I can usually find one empty box, or maybe one almost empty that I can move the products from into another box for the same product next to it. Then I just take the box and use it to transport my groceries. Stores just throw those boxes out anyway, so they don’t care if you take them (I have asked). At this point it’s a bit of a game for me, to guarantee I always find a box. I have a personal rule never do anything that would make the lives of the workers harder in the process.
Another thing you can do is just take a cardboard box from some product in the store. This may depend on country, but where I live the shops leave products on their transport boxes on the shelves. Walking around the store I can usually find one empty box, or maybe one almost empty that I can move the products from into another box for the same product next to it. Then I just take the box and use it to transport my groceries. Stores just throw those boxes out anyway, so they don’t care if you take them (I have asked). At this point it’s a bit of a game for me, to guarantee I always find a box. I have a personal rule never do anything that would make the lives of the workers harder in the process.