Of course the details will vary with locality. Many gites/auberges are open seasonally; in MMD one sees the phrase "Paques a la Toussaint" (Easter to All Saint's) very frequently. For some, that will include the week before Easter. For others, it means they might be open as early as Easter, some days each week, if someone has made a reservation. The contact info for almost every lodging within a mile either site of the GR65 is given in MMD. So get a copy and start sending emails.
Since even normal weekends in France usually involve Sunday-Monday closures, and the major holidays in May frequently become four-day (Thursday through Monday) closures, Easter looms as something of a disaster for the unsuspecting traveler. While it should be possible to find some businesses open in the larger towns and cities (Le Puy, Figeac, Moissac, Condom), in the small towns and villages one should not expect to find anything.
Be prepared by shopping in advance. When you make your lodging reservations, check with your host on the local situation and ask for suggestions.