We're only staying on the French side for our first two nights, and the information that I got from GCBC was that we only needed to present our vaccine cards; however, that was before France dropped the US from the "green" to the "orange" category. I sent an email to GCBC last evening just to confirm that nothing has changed, and haven't yet heard back from them.

However, I just found this information on the US Embassy site, which confirms that fully-vaccinated travelers to France from the US need only present their vaccine cards.

"Travelers to France:

Fully vaccinated travelers* face no restrictions on travel. Such travelers need to present their proof of vaccination (French government officials accept the CDC card for this purpose) and a sworn statement (normally provided by the airline) certifying the absence of COVID-19 symptoms and of any contact with a confirmed case of COVID-19.
* France considers a traveler to be fully vaccinated two (2) weeks after the second dose of the Pfizer, Moderna, or AstraZeneca vaccines, and four weeks after the single dose of the Johnson&Johnson/Jansen vaccine. No other vaccines are acceptable."

Here is a link to the site if you want to read the information for yourself:

Link

Edited to add that I heard back from GCBC, and they told me to check with my airline, lol. So....I'm going with the assumption that the proof of vaccination will be sufficient.