We’ve had a rule like this in our county for several years now. I think I’ve paid for maybe three bags in that whole time. You get used to bringing reusable bags.
I suspect that most of the objections to this are nonsense put out by the National Plastic Bag Council or something like that. I don’t think any raw food has ever touched the inside of one of our reusable bags, except perhaps for the peel of a banana.
Edited to add: It turns out it wasn’t the National Plastic Bag Council. See: http://mediamatters.org/research/2014/10/08/californias-plastic-bag-ban-myths-and-facts/201064
Media Matters is a partisan site, but it provides links to the studies on which it relies.