As IPA, if you wish to buy a car in France, the only possible registration is a French standard registration. You are not eligible for the tax-free purchase or the diplomatic registration of the car.

PROCESS FOR A FRENCH STANDARD REGISTRATION

If you buy a new car from a French car dealer, the dealer will register the car for you and you will receive the car registration card directly.

For a second-hand car, you will find information sheets below according to the case :  

Register a second-hand car bought from a French car dealer

Register a second-hand car bought from individual in France