Technically there is no such thing as a "Family Coat of Arms", not in Europe at least. The Arms were individual identification and inheritable property so only the one who inherits from his father is allowed to bear that design, and only after the previous bearer has died, all other members must submit a request for their own designs, usually the same basic design with a small difference. And not all descendants submitted requests. So one given family branch or even line can have dozens of designs over the centuries.

Having said this, there are some useful resources that you can check. They are all books and most of these can be found in the online library Internet Archive, what you can't find there you can probably find in the Bibliothèque Nationale de France (BnF) who has a very large collection digitized and available online free of charge (for non-commercial purposes only), through their GALLICA page. Below is a list of publications which I have found to be extremely useful to me, most are series containing several volumes:

ARMORIAL GÉNÉRAL DE FRANCE. There are several editions over the decades, the largest is by d'Hozier but there are also editions by M. Borel, Firmin Didot and G.A. Prevost. The content varies a bit from edition to edition but basically contains the person's name, a B&W illustration of the arms, the date when the arms were first granted, the genealogy submitted by the person or his ancestor to prove his claim to nobility and the blazon (written description of the design). The genealogy will contain mostly the male line which had rights to those arms (ergo, usually of the eldest son or next in line if the eldest died before his father).

GRAND ARMORIAL DE FRANCE. Also a d'Hozier series. Look first for the 1865 edition which contains an Index of the series, it is most helpful and can save a lot of time.

There are also dictionaries and guides explaining devices, figures, colors, etc.

If you manage to get far enough back in your family history to when your ancestors were still Catholic, if they were already members of the nobility that early on, chances are that some may have been Knights in some Chivalric Religious Order such as the Templars and Hospitallers/Malta (the name changed later in time when they changed their home-base from the Holy Land to Malta). These orders also kept genealogies and arms designs of their members on file. If you find that you have one or more ancestors belonging to these orders, you will probably also discover that the families tended to band together and marry each other.

Hope this helps a bit!