France's outdoor markets — the weekly marché that's a genuine fixture of daily life in most towns — are generally relaxed about a well-behaved, leashed dog walking through with you, the same way any pedestrian street would be. The stalls themselves are a different matter: individual vendors, particularly those selling fresh food directly, can and do ask a dog to keep its distance from the produce on hygiene grounds.
Stepping inside a small indoor food shop — a boulangerie, fromagerie or boucherie — follows the same logic as a restaurant's dining room: there's no blanket national rule, and it's the shop owner's call. Many small shopkeepers wave a calm dog through without comment; others, particularly where space is tight behind a counter, prefer the dog waits outside.
Supermarkets are more consistently dog-free indoors, following the same food-hygiene logic that keeps animals out of a restaurant's kitchen, though a small dog in a carrier is sometimes an exception depending on the specific store's own policy.