Looking for a place to study French and stay as a family for the month of June

I would begin by finding where the French studies that your institution will approve are to be found. That will be your first list to look at. Come back to the forum with some of the cities/towns/areas that you found. France is a big country (about the size of California) and your question is a bit too broad. IMO, almost any place in France is worth staying in for 2-3 weeks, and longer, and many, many, many offer parks/museum/cultural things to do.

First, French children will be in school in June, so there won't be vacation activities for children except on Wednesday afternoons and Saturday afternoons, times most schools are closed . For that reason, a Paris school for you would probably be best, so your family has access to the most activities and day trips. There should be rentals available in close-in suburbs that could fit your needs.

I'm not at all up-to-date on language schools in Paris, nowadays, but agree about asking your institution which schools they work with, and checking AAUP resources.

What sort of French lessons are you looking for? Do you have any French knowledge already or are you a beginner? Do you seriously seek in 2-3 weeks enough new ability to be useful to your academic career (and if so, is your focus more on conversational French, or in being able to read it?) or (meaning no offense) is this more of a casual endeavor along the lines of a paid vacation? If the former, then leave the family at home and find a total immersion program where you will live with French speakers and have to speak French and only French every minute of every day whether in or out of class, and preferably with a longer stay. Focus on the quality of the program rather than where it is, in fact if it’s in a place without a lot of tourists or distractions from your study, so much the better. If the latter, just pick any French city in an area that appeals to you and your family and there likely will be some place to study French there, whether a school or a private tutor.

Thank you! Yes, at the moment we are broadly searching, and I have already started taking lessons, but looking to have a more intensive workshop of sorts. Right now, I am in conversation with a couple of schools that have struck a good chord, one in Antibes (more corporate feeling) and another in Fabrezan (more customizable, very local/intimate). We understand that in Fabrezan we would likely need a rental car, which brings to another question, how recommendable is driving for tourists? Is it "easy"? Merci!

Fabrezan has a population of 1,254. Not a good idea to make children stay for a month in such a dead place. My in-law's village in Burgundy was about that size but my children visited their grandparents when the other children were out of school or they were enrolled in the village school. But after spending many summers there as an adult, it gets very boring.

If you are looking to come south, you could look at the schools in Montpellier. I meet Americans, including university people, who come to Montpellier for language instruction. The only problem, as I said before, is that French children are in school, so your children will really be alone. This happened to my granddaughter, so now we shoot for summers so she can be with children her own age.

Yes, in Fabrezan you will need a car. Antibes is a bit better. Having had children and grandchildren in different areas of France, I suggest you look at Paris or one of its suburbs so there is plenty to offer the children, as I said before.

Between those two cities as somewhere to stay for 2-3 weeks, obviously Antibes is the logical choice. OTOH, if you found a place to stay in Fabrezan or nearby, you would very likely get a lot more language practice in the local shops, cafes and markets. Your husband would definitely want a car to take the kids out exploring nearby Carcassonne, as well as the coast. The cost of the car might be offset by the (likely) lower cost of accommodation. Driving outside of cities is very easy. If dropping you off at school in the morning and taking the kids somewhere during the day appeals to your husband, take a look and see what is available. There are two sites for gites (French for country vacation house), and they both have very straightforward names like gites-en-France.com. Of course, the usual sites work, too. Bon chance!

Gerry's suggestion of finding a gite to stay in is good. The website is gites-de-france, not "en" France. We have used them many times through the years, for short and longer stays, from 2 to 7 people, from chambres d'hote to large houses. You will need to have an area or town picked out before looking. So narrow down the search for that language school first
Driving in France (outside of Paris and Marseille and Bordeaux....traffic there, yikes!) is pretty simple. Just learn how to read the signs and the basic road rules, and do not speed! The cameras that record your speed will get you a ticket that may arrive a few months after you get home. Get a Michelin paper map for your area. They give you a better overall picture of the place than a Google map or its like. Plus it makes a good souvenir!
We took our kids to France when they were 10 and 8. Also when 20 and 18, and in their forties, along with grandkids of 9 and 12 years. All this was in the summer. The grandkids were in early June, so it was when French schools were still in session. With the grandkids it was the Dordogne region, and they were never bored (castles, prehistoric caves, museums, markets, chocolaterie, kayaking, even grocery shopping!) Our gite had a pool and large grounds to explore, so they weren't bored when not sightseeing. (They also each have kindles, so down time reading was a daily unwinding ritual). They also helped with the cooking and the laundry and the everyday things, so the house we stayed in became "home" to them while we were there.
I think the small town of Fabrezan sounds a bit too isolated. Find someplace where the kids and Dad can wander around on foot, rather than having to leave a place every time they want to see or do something. A coastal city, whether in the Riviera or in Normandy or southwest France could be fun exploring the beaches (not necessarily swimming). Our grandkids love visiting tide pools, for example. Just a thought.

Check out this website. It was started by a woman that also does podcasts and tours. She does a French Language "boot camp" every year, so perhaps she might be a resource. Also, look through her podcasts, there have been guests who have discussed their experiences immersing themselves into France for a more extended time frame.

https://joinusinfrance.com/