Are you planning to walk from Lyon, along the route that joins the Cluny route before getting to Le Puy? There are some steep rocky sections uphill where you join the Cluny route, from St-Georges-Haute-Ville to La-Chapelle-en-Lafaye, where the altitude change is about 400m to just over 1000m that might be difficult. There are also some steep descents early on the route from Le Puy that could be tricky in the wet, eg down to St Privat d'Allier, and downhill from the chapel at Rochegude (which I found hard enough over big rocks with short legs!) There are other flatter sections, eg over part of the Aubrac Plateau, where the chemin is quite rocky/stony that I imagine could be uncomfortable with a cart. But otherwise, I imagine it would be perfectly possible.
Margaret