This is a challenge itself, you have to be really determined to stick to your diet.
Okay, maybe for a simple vegetarian it is easier, because you still have option with eggs and dairy products and usually these are the base of the vegetarian version. I am a quasi vegan - a vegetarian with lactose intolerance. Yeah, I know, it is not an easy task for someone who cooked the last 20 years the traditional meals. So my expectations are not high, even though I can be surprised. For example with this:
That was not at OB, but in Romania at a kindergarten in Cluj-Napoca, where I was volunteering too. I think it express perfectly the problem. Usually the kitchen staff just don't understand what does it mean "don't eat meat", and they offer broth because there is no meat in it.
The magic word which is works is 'post'. It means fasting in Romanian. During the year the orthodox believers are many times on fasting food, avoiding all food of animal origin.
So you can ask for 'post mâncare' / 'postos étel', and you won't be disappointed. Even at the restaurants, in the shops you can find this 'post' labelled foods. Remember it!:)
To buy the vegetarian/vegan ingredients for cooking I recommend the local market (near to Penny), or Elikson for vegan milk. Everything else, what you can't find in Sovata, probably you can buy in Targu Mures at Kaufland. If you are going to the city, bring with yourself some food containers and you can buy aliments without packaging in a little shop, 'New Life Market' (Str. Aurel Filimon nr. 34).
In summary you should consider this as an opportunity to improve your creativity and assertiveness.
It is not impossible but challenging. :)