You can always get lucky with accommodations made in person on arrival. However, in my experience, it is less stressful to have a reservation waiting for you. I always make one for the first two nights at SJPdP.
There are numerous private albergues, hostals and hotels to reserve in advance. Try
www.booking.com.
Also, if you are on the noon train from Gare Montparnasse, you will not arrive in SJPdP until about 6:00 pm. A previously made reservation is the recommended way to go. Get checked in. Have a great trout dinner in a local restaurant with fish fresh caught in the local river. Relax...
I use the intervening day to play tourist and finalize my packing, shipping stuff ahead (Express Bourricot), and making the necessary visit to the Pilgrim Office at Rue de Cittadele #39. That evening, I attend the marvelous Pilgrim Mass at the local church (ALL are absolutely welcomed). Then, the following morning, I start my pilgrimage towards Santiago.
That first day gives UP an entirely new dimension...when Refuge Orisson appears around that last bend in the road, it is like an oasis in a desert and the cold beer tastes all the better. You will KNOW that you earned it.
St. Jean Pied de Port has been supplying and sending pilgrims off, on their way to Santiago for more than 1100 years. They are seriously good at hospitality.
The recommendation to book ahead is especially true if you are walking during the "official" Camino season, from Easter through September. FYI, in 2018, Easter Sunday falls on Sunday 1 April (NO, not an April Fools Day joke).
I hope this helps.