The routes in France are never as busy as those in Spain. They intend to get busier the closer you get to the Spanish border. The Via Lemovicensis is about 900 km from Vezelay to St Jean Pied-de-Port and if you are walking 2 weeks from Vezelay I doubt you'll meet any crowds.
The CSJ website has this to say of accommodation:
More and more people are becoming aware of the difficulties encountered by pilgrims. This is due to the fact that the public authorities and the people along the route are more and more conscious that accommodation is inseparable from the welcome extended to visitors, which is generally quite good. All the way along the route itself or nearby there is a varied selection of accommodation for the pilgrim, beginning with the pilgrim refuges (the number of which is increasing steadily), not to mention the town halls (who can generally find a solution, however basic, to lodge the needy pilgrim for the night), small or large country hotels, chambres d’hôte (bed & breakfasts), or gites d’étape (fairly numerous in certain regions), and even numerous rooms for pilgrims provided by generous & charitable private individuals.
You should be just fine!