Karen,
As Sasha mentioned, it's quite likely that La Torretta won't be open until April 1st, as that's the "official" start of tourist season. That could well explain why no vacancies are listed on their website.
Are you firm on staying in Manarola? There may be a few places open there in March, but you'll have a better chance of finding something in Monterosso. It's the largest of the five towns and therefore has the greatest number of hotels, restaurants and other tourist facilities. I don't know how current THIS LIST is, but you could check some of the lodgings to find out when they're opening for the season.
One good possibility would be Hotel Pasquale, which is a wonderful hotel in the old town area of Monterosso, and an easy walk from the station. Their website indicates they'll be open on 1 March, but you'll have to contact them to see if they have vacancies. I usually stay at their sister property, Hotel Villa Steno, but not sure when they're opening?
Happy travels!