Yes, you can camp in Paris itself. Not only that, you can get a campsite overlooking the Seine River in the midst of the vast, forest-like Bois du Boulogne city park.
Camping de Paris - Bois de Boulogne is the only camground within the péripherique ring road of Paris.
Happily, it's also a four-star campground with grass and shade trees around 510 sites spread over 17 acres — including 50 pitches for campers with a tent and no car. It is located in the 16eme arrondissement, on the Left Bank across from the southern tip of the Île de Puteaux.
The campground offers heated restrooms, a supermarket, bike rentals, laundry service, a playground, and a bar/restaurant selling pizza and other fast food.
You can also rent tents (complete with sleeping bags), cottages (sleeping up to six people), or wooden gipsy caravans (sleeping up to 4 people).
This Paris campground is open year-round.
To get to Camping de Paris - Bois de Boulogne: Take the Metro line 1 to the "Porte Maillot" stop and walk out the André Maurois exit. Hop on bus 244 through the park and get off at "Les Moulins Camping." There is also a private shuttle between the Metro station and the camp office. Total travel time: about 15 minutes.