It's highly doubtful that you will find any apartments listing vacancies this far in advance, and the short-term rental laws are changing. By the time you arrive in Paris, the vacation rental laws will probably have evolved even further, and it might be impossible to find an apartment in Paris - especially given the size of your family. However, in the nearby suburbs, where there are virtually no restrictions on vacation rentals, you could conceivably rent an entire house or multiple apartments with little trouble. It won't take long to get into Paris proper, and it might be the easiest solution.
Otherwise, you could book a couple of family rooms in legal apart'hotels such as Citadines or Adagio, which have kitchenettes and laundry facilities. Both chains have multiple locations in Paris, price varies by neighborhood, and discounts usually apply to advance bookings.
You won't have any trouble renting a "gite" in the countryside. You'll have to make some decisions about which activities matter most to your family, before deciding on a home base. Do you want to be near the beach, mountains, museums, ancient ruins...? Not all of that is usually available in one location. If you do some research using "Tourisme ______" and choose a region, you'll get lots of ideas of what there is to do in the area.
You can take the TGV to the South of France, and benefit from discounts since you have multiple passengers. After that, if you settle in a major city, transportation will be easy.
Most of the time, there will be bus and train service in rural areas, but not always where you might want to go. A couple of rental cars might come in handy from time-to-time.
