If you are going to do it make sure you pick a large, lively site where you don't need to go off site for anything. Honestly, having been on quite a few Eurocamp type holidays I really can't see how you can make it work without a car, but then I like to go out and about every day.
The on-site supermarkets are expensive (although some are quite large and well stocked, it really depends on the size of the campsite) so even getting in enough shopping for a week's self catering will be very tricky and they tend to have only one restaurant on site, which can get boring. Most of them are not placed right near beaches or major attractions although some do offer shuttle services or are near bus routes.
Given they are not especially cheap and you'll still be paying out for flights to get there I think it might be a false economy and you'd be better off with a small basic hotel somewhere fairly central in a resort with everything to hand.
That said, if you do do it, and are happy to chill around the caravan, let the kids play in the pool and run freely around the grounds without any huge expectations re: sightseeing, and if you don't mind eating from a limited menu or cooking with food from a limited range in the shop you will still have a lovely time.
Look at Dol-de-Bretagne, Biscarosse, (sorry can't remember exact names of the sites) Norcenni Girasole in Tuscany, La Sirene in Argeles, and any others that are similarly large and not too far off from the airport as you'll have to get there by bus or taxi.
And if you are going anywhere (with the exception of northern France) in the height of summer then don't even think about booking one without air conditioning. It will be hell.