The way from Le Puy is great for camping. Official campsites might be closed that late in the year, though.
I've got a list of campsites along the way from Le Puy, it's in another thread. I try to find the link. You could then check the websites of the campsites whether they're still open in october or not. But I wouldn't be too optimistic - camping season is short in France, and many campsites are only open from may or even june-september, only few april-october. I'm pretty sure most will be closed in november.
You might have more luck camping in the gite gardens. Many allow it for a fee (usually 8-10 Euros), and that includes use of bathroom and kitchen in the gite, and you'll also meet other pilgrims there
The miam miam dodo should include the information whether a gite allows tents or not.
If you plan to wild camp in a respectful way - I think it's not exactly legal in France, but different to Spain it is usually tolerated as long as you do follow the basic rules (set up camp late, leave again early, no fire, carry out your waste, basically "leave no trace and don't bother locals, wildlife or farm animals").
There are many places, often next to picnic stops, that work well for wild camping. I've even seen a spot in a small town that explicitly said something like "wild camping for pilgrims allowed here, from 7pm-7am for one night".
There are also a few official, very basic shelters that are a nice alternative to sleeping in the tent. I remember at least two (refuge de la béate and one after Ostabat, I think).
There are not many pilgrims on this forum who like to camp, but I'm sure many will be able to give information regarding the weather that time of the year.
I haven't walked the route that time of the year, so can't say anything about temperatures ect. then. But even in may it was pretty cold some nights, so you definitely need adequate gear, especially a warm enough sleeping bag.
Happy planning and bon chemin
Edit: Link to campsite list
Also, I remember I walked the Aragones from Somport in late october/beginning of November. It's the other side of the pyrenees, but shouldn't be too different on the other side and give an idea what to expect for the last part of your walk. Weather was sunny and warm for the first days (~20°C) then cold, windy, constant rain for the rest of the time (~2°C and snowy rain). So I guess it's difficult to predict and will probably be different mid-october and beginning of november.