This seems to be an inaccurate statement that will create rumours and unfounded alarm for many people.
How did you search?
I have just used booking.com (which we know has only a fraction of the accommodation available) and I checked all 30 days in September - one by one - to look for accommodation. There was albergue lodging available on every single day, except for two Saturdays - the 17th and 24th. On those days there was other reasonably-priced accommodation, but admittedly quite limited selection there on booking.com.
What exactly did they say? That they were filled every day from May 1 through September 30? I don't know how recently the
website of Pamplona turismo has been updated, but it says something quite different. This is something that could cause confusion in translation but on the website, the English and Spanish versions are consistent:
Bookings. The hostel offers a booking service for pilgrims and tourists alike, except for the period from 1st May to 30th September, both inclusive. During this period, no prior booking will be accepted and the places available will be occupied on a first-come first-served basis.