I walked the VF from Lucca to Rome in May 2024 and saw pilgrims every day. However, when I asked several different hosts in various places about the timing of pilgrim traffic, most said they don't get much traffic after September or early October. This is mostly because the Italians are back to work and kids are back in school, so there are fewer pilgrims on the VF (and it is already a lightly traveled path). You should have little trouble finding accommodations, as most of the places I stayed were not albergues or hostels, but were AirBNB or private rooms or small hotels which are open all year. The list of accommodations from the Confraternity of Pilgrims to Rome was quite useful, as was the handbook by Sandy Brown and of course, Booking.com.
Buen Camino