No. you don't need to camp..... but pre-booking is the norm on the Le Puy route, even just a day or two ahead. Groups of French people who are walking on the route, whether as pilgrims or 'randonneurs', have sometimes booked months in advance. If you don't book you will find something, but you might end up in a chambre d'hote (like a bed and breakfast) and end up paying quite a lot more. The Miam Miam Dodo guide (available from CSJ) lists all accommodation available to pilgrims, with prices etc.
We found it stressful in the latter part of May, from Aire-sur-l'Adour south, as we happened to coincide with a large group from Dijon, with two vans and 17 people. (If you have walked in Spain, 17 may not sound like many, but in France, 17 could book out a whole gite sometimes.) We could have solved the problem by waiting a day for them to out-walk us...but by that stage we were keen to push on to SJPP. But while we shared the route with them, we often found that our 'first choice' of accommodation was already full, and we needed to think laterally. Vehicle-supported groups in France are quite normal - and they will nearly always have pre-booked.
If you don't speak French, the tourist offices in France are widespread and will usually be willing to book ahead for you. Alternatively, I met someone who took a phone with her Miam Miam Dodo guide (which is carried by almost everyone seeking accommodation on the Le Puy route) and she always found fellow French-speaking walkers who were willing to use her phone to book ahead for her.
Margaret