Too bad I didn't notice this topic earlier, could have recommended some places.
If you recommend some places on this thread now, the next person who searches for where to eat in Helsinki will be able to search this and be grateful for your help!
True! So, there's only a few completely vegetarian restaurants in the city but eating vegan is relatively easy.
All the chinese places I've visited have tofu dishes that are vegan if you order them without fish sauce.
There's one pizza place in the Kallio district that has soy cheese: Restaurant Salpimienta in Fleminginkatu 7.
In the same part of the city there's a few vegetarian restaurants. At Soi Soi (address is Vaasankatu 9) you can get fast food like seitan kebab and tofu burgers.
Then there's Silvoplee (address is Toinen Linja 3) which I've never visited but I've heard the food is awesome. They also have organic beer and wine.
Veganissimo is an all vegan lunch restaurant (open from 11 to 16) in Kulmavuorenkatu 2.
These four places are about a 5 minute metro ride, plus max. 10 minutes of walking, away from the city center.
Downtown there's Vegemesta, another place with vegan/vegetarian fast food. The address is Pieni Roobertinkatu 2-4, they also have a food truck every friday and saturday at Fredrikinkatu 44 (next to the exit of Kamppi metro station). It's open from 19 to 4:20 so definitely a good spot to get some munchies after a night at the bar :)