It’s very tough being a true vegetarian in Tokyo. That is not my case, but it is for some of our clients and friends who come to travel, or live and work in Japan. Here are some suggestions for restaurants that are vegetable-friendly. They may not be strictly vegetarian, so be sure to inquire if you follow a very strict diet.
When I am craving vegetables, I make a beeline for Rose Bakery and get the salad lunch. There is a branch in Ginza, but you most likely will find me at the Kichijoji branch.
Sougo is a lovely vegetarian restaurant in Roppongi.
Daisuke Nomura is the owner. His family owns another famous restaurant called Daigo in Atago, also Buddhist vegetarian.
I LOVE Dhaba Indian in Kyobashi for dosa. It is a short walk from Tokyo Station:
I often ask for vegetarian tempura at places like Tenmatsu (there is also a branch in Shibuya):
Kushiage are skewers of vegetables, fish, and meat, that are breaded and deep-fried. Hageten in Ginza said that if you call ahead when making a reservation that they could do a vegetable only flight of skewers.
This is my favorite pizza in Tokyo:
If you like bagels, this shop is also nearby the pizzeria above:
Also, I LOVE TY Harbor restaurants as they are very good about substituting vegetarian dishes for meat or seafood components.
The following are all in the TY Harbor group:
Aoyama Cicada
Daikanyama Ivy Place
Sukiyaki is a dish based on wagyū beef, but if you call ahead, they can prepare a vegetarian sukiyaki.
For Japanese breakfast, the Park Hyatt Tokyo’s Girandole will also do a vegetarian traditional Japanese breakfast, but reservations need to be made in advance for this.
Some other spots from my blog:
Food Sake Tokyo 
A friend came to visit us in Tokyo who had been a vegetarian since she was 11. She did ok when we were with her because we could explain that “no, she can’t eat shrimp; no, she can’t eat bonito flakes; no, garlic is not actually meat (even though the name implies that it is) and she can eat garlic.” But when she and her husband went to Kyoto without us, she gave up because she was just too hungry. She started eating meat and hasn’t looked back since! We always joke that Japan breaks vegetarians 😉
Great story! My girlfriend was here for a few years and eventually gave in to things like soba, which has some fish in the dipping sauce. But, once she went back to the US she was back in her groove. 🙂