Generally speaking, the bag-transport services are your best bet on the Le Puy route. There is bag service along the entire Le Puy-SJPP route, provided by different firms in different areas. Your gite host will be happy to call and make the arrangements. This will involve waiting at the gite for pickup, which might not be until afternoon. There is not, generally speaking, taxi service available to pick you up any time and any where. If you are in or near a larger town, it may be possible (if not a Sunday, Monday, or other local closing day).
And - speaking of running on the Le Puy route - have you seen the size of the cobblestones? There is quite a good percentage of the route where, even in stiff-soled hiking boots, one must be very careful about placing one's feet.