I can't think of any reason why you couldn't do this... Le-Puy-en-Velay has a long history of Marian pilgrimage to it, quite separately from any connection with St Jacques. The people running the gîtes might be a little surprised to see you are going in the reverse direction, but I can't imagine any would turn you down. In France there are people on various walks who use the gîtes, and by no means all of them are walking a 'pilgrimage'. Many are just walking for a week or long weekend.
The FFGR French Federation of Long Distance Walkers is much more focused on marking walking routes for recreation than for pilgrimage, so routes are marked in both directions. The map sketches in MMD are graphical, so their is no directional preference there (the German and French walking guides are described in the west-bound direction). The landscape is up and down whether you're pointing east or west; heading east, though, the general trend is up. Also, there are some slopes that will be steeper going that way.
Roch, depending on when you plan to walk, you just need to be aware that camping grounds are often 'seasonal' in France, and may only be 'opened' as such for a few months a year. Some of this walk is also at quite a high altitude-1300m or so- eg across the Aubrac Plateau, but also near Montbonnet and Le Sauvage, where the weather might not really be easily conducive to camping.
Most presumably will be, but I know from when I cycled in the Loire, staying in camping grounds,that opening times varied tremendously. A few are open all year- others start opening in May/June, and a few are only open for the main French holiday period in July-August. Local tourist offices have more exact opening dates for their own areas.