A summary of USDA requirements for pets to travel from the USA to France can be found here. Basically, your pet is required to have the following:
1. registered microchip and successful scanning.
2. rabies vaccine only after the microchip has been scanned/activated AND a minimum of 21 days before travel.
3. licensed veterinarian completes a non-commercial health certificate and declaration.
4. share the health certificate with USDA and get an endorsement signature within ten days of arrival (there is an online submission option as well as a mail-in option). The French government wants the mailed option, which provides an ink signature and an emboss mark. This is a required document for and during travel.

Also see:

https://www.aphis.usda.gov/aphis/pet-travel/service-centers-endorsement-offices