Possible. I walked from Santarem to Porto in late Oct and stayed in alberues apart from Tomar. Santaren: Casa de Misericordia, double rooms for the price of dormitory, centre of town. Golega: Solo Duro, nice location, friendly dogs. Tomar: good hostel 2300 Tomar, centre of town. Alvaiazere: Albergaria Pinheiro - small one, rooms for 2 people or private ones, staicase and stamps to rememer
Rabacal: go to the private one, at Cafe Bonito, brand new, have a swimming pool. Coimbra: Rainha Santa Isabel, before crossing the river to the city, huge, old and atmospheric. Mealhada: Alb Hilario, check if they have hot water in the dormitories, we didn't but the ones staying in private rooms did. Agueda: probably the best albergue on the Portuguese way, Alb St Antonio. Slightly out of town centre but on the way for next day. Albergaria: we went to A-Nova - avoid as almost on the highway, very dusty and polluted. Stay in A-Velha instead, greener and more pleasant town. Sao Joao da Madeira: we stayed in Casa da Misericordia, donation, not exactly an albergue, huge room, mattresses on the floor. People with whom I ended up walking later went from Lisbon and stayed in albergues as well. Hope that helps.