is there luggage storage at the train station?
Not as far as I'm aware, ad I would be very surprised indeed if anything existed at the station. Levanto is not listed as having any luggage storage on the official railway website:
Luggage storage at railway stations in Italy (and other European countries) is much less common than in the past because of security concerns. Italian railway storage facilities are always staffed, for the same reason.
There is a staffed left luggage facility at La Spezia Centrale station, not too far away (from 20 minutes on the train, and trains are frequent) so perhaps you might use that?
Yes there is a service that provides luggage storage services all over the city, Railway station included. You can find the closest luggage storage here
That link is for a service in Rome. The question is about Levanto.
If you are going to promote a website it's best to make sure it's relevant.