This question is very broad and I'm not sure whether you mean to restrict it to dishes you'll be able to buy in restaurants or whether you will also be cooking meals yourself using locally bought ingredients. But you say you want to eat local food which is vegetarian and you have no idea where to start, so here is some information about the main flavour bases used in French cuisine and in the cuisines of several other European countries.
Although unfortunately the proportion of the population in France who are vegetarian isn't high, the good news is that the flavour base of French cuisine is vegetarian. (Don't let anybody tell you otherwise!) Not only that, but so are the main flavour bases in every other country or culture I have looked at. In Europe these include Italy, Spain, Portugal, Germany, Poland, and Greece. (I haven't investigated Belgium.)
In France the flavour base is called mirepoix. This is made from onion, celery, and carrot, usually in the ratio 2:1:1. These vegetables are chopped and slowly fried.
The cutting of vegetables in French cuisine can be to various different shapes and sizes, and many of the cuts have their own specific names. Julienning is one that is well known among anglophones, but there are many others. You can read about culinary knife cuts here.
Examples of flavour bases, with their names:
France: onion, celery, carrot (mirepoix);
Portugal: onion, green pepper, tomato, garlic (refogado);
Spain: onion, red pepper, tomato, garlic (sofrito);
Germany: leek, celeriac, carrot (suppengrün);
Poland: leek, celeriac, carrot, parsley root (włoszczyzna);
Italy: onion, celery (or fennel), carrot (soffritto);
Cajun culture in North America: onion, celery, and green pepper (the Holy Trinity);
Greece: lemon juice, olive oil, oregano.
Summary of European or European-influenced bases, not including the Greek one:
- onion or leek
- celery (or celeriac) or tomato
- carrot or pepper
herbs and spices
- perhaps parsley; in France and Italy, also bayleaf
- perhaps garlic; in Spain and in Cajun culture, also paprika
I can recommend two books by Paola Gavin, namely Mediterranean Vegetarian Cooking and French Vegetarian Cookery, and also Cooking with Flavor Bombs by Giovannina Bellino.