TheFork users who ate in Portovenere have said really good things about Il Timone, L'Anciua and Pizzeria I 3 Scalini. Maybe you should give them a try!
Some of the more popular pizzeria restaurants that offer vegetarian options on their menu in Portovenere according to TheFork users include:
Il Timone, with a 8.1 rating
L'Anciua, with a 8.1 rating
Pizzeria I 3 Scalini, with a 7.9 rating