Welcome, mija! There's a couple of us on the Asian side who are either vegetarian or generally prefer to eat as such and we'd be happy to host you on a veg-friendly night in Kadıköy.
Çiya in Kadıköy is great for vegetarians. A cold starters bar with tons of vegetarian options (dolma, hummous, bulgur köftesi, salads, etc), a vegetarian soup, and a few daily vegetarian entrees. In addition, they do a meatless kebap.
Some other finds for vegetarians: 'etsiz çiğ köfte' which is bulgur-based, spicey, and served with lettuce, parsley & lemon juice. It is cheap (2.5 ytl per dürüm) and delicious. One of my favorites, Komagene, has branches throughout the city:
http://www.komagene.com/default.asp?m_i ... e=1&title=Şubelerimiz
Ironically, some of the best vegtarian dishes can be had at your standard kebap restaurants. They are usually more than happy to make a vegetarian kebap for you which is a blend of fire-roasted vegetables such as eggplant, peppers, tomatoes, onions, etc. Hamdi in Eminönü across from the otogar is a good example.
Afiyet olsun!