Seems unlikely that they would let civilians walk around with rifles and shotguns to go hunting. Was all hunting banned, even for commercial purposes?
So France was divided into two zones the Occupied Zone in the North and the Zone Libre in the South (Vichy France). In the Occupied Zone all forms of hunting were banned outright. The people didn't have ready access to firearms and ammunition anyways, so even if it had not been banned the people wouldn't have been able to really hunt anyways. Now hunting wasn't the only pastime to be repressed. Raising carrier pigeons was banned (it was a popular activity in North Eastern France) and the French film industry while not out right repressed was dealt a blow when the Germans founded their own French film company known as "The Continental Film Company"; it had more resources than any other French film company and was thus able to basically out produce and out bid all the other French film companies.
Fishing, however, was not banned and quickly became a very popular activity. Fishing stores and markets in Paris sold record numbers of bait, tackle, and fishing poles. Every morning people would flock to the Metros in order to get a good spot along the Seine river for fishing. There is a story about French writer Jean Guéhenno, who one day got up and went to the Metro at 5 AM trying to get a train, only to find it was already clogged with fishermen.
The go to book on the occupation of France is Julian Jackson's "The Dark Years: France 1940-1944"