If you were willing to pay approx. 8–9 euros for lunch, there would be lots of good options. For example:
- Satkar Kamppi (Nepalese restaurant, tripadvisor, home page): vegetarian options in the lunch menu for 8.40 euros.
Note that the quoted prices in the lunch menus are precisely what you will pay (assuming you drink water). There are no additional taxes or service fees, and you don't add a tip in Finland. You can try to use eat.fi to find good restaurants in Helsinki (you can, e.g., search for "vegetarian friendly" restaurants with high ratings that are open right now).
Your budget of 6–8 euros will be a bit more tricky (other than self-catering). Some suggestions in this price range:
Armas Pikaruokala (tripadvisor, home page): very good vegetarian lunch ("kasviskebab") for 8 euros.
Student cafeterias (home page): lunch starting from 7 euros (cheaper if you have some connection with the university), healthy (not necessarily delicious), and there are vegetarian options. I think one of the cafeterias ("Ylioppilasaukio", in the middle of downtown Helsinki) also serves lunch on Saturdays.
For very simple self-catering, buy a vegetarian Thai cube from a local supermarket. You will need only a microwave oven and a fork. The price should be less than 4 euros.
For weekends, a reasonable compromise in this price range might be to buy some ready-made meals from Stockmann (a large department store in downtown Helsinki). Downstairs they have a supermarket ("Stockmann Herkku") with a good selection of all kinds of ready-made meals; you will certainly find some vegetarian options there as well. In addition to the pre-packaged options, you can pay by weight (just go to the service counter and make your order), so it is possible to put together a relatively inexpensive and diverse meal by taking a little bit of this and that. They are open also on Sundays (except on public holidays). Stockmann is definitely not the cheapest grocery store in the city, but the selection and quality is good, and it shouldn't be any problem to get service in English. There might also be a possibility of having your lunch there if you don't have a microwave oven (I don't have any first-hand experience of that, though).