Tenpura ryori, Yasai Tendon, Yasai Itame (sauteed veg), Yasai Yakisoba (veg noodles) and many other veg options can be tried out

Tenpura Ryori - Tenpura is a Japanese dish which is fried in a batter. Vegetable (yasai) Tenpura would be vegetables dipped in the batter and then fried. They are assorted vegetables and then as a side dish rice or miso is given with it in a full meal.
Yasai Tendon - Basically a bowl of rice with toppings of Vegetable Tenpura (while ordering ask for yasai or veg ONLY else sometimes few of these tenpuras are of prawns etc).
Yasai Itame - Sauteed vegetables - Many small izakaya or shops will have this - basically vegetables sauteed in soya sauce with little bit of salt etc.
Jyaga butter - Jyaga stands for Jyagaimo which means potato -so basically it is a hot potato with salt and butter on top of it. Tastes very good ...
Yasai Yakisoba - Noodles with vegetables - these are dry noodles with no soup. (while ordering again ask for Yasai (vegetables) only.

Okonomiyaki (kind of pancake) is also very delicious. While you can remove all meat etc, egg might be mixed with the floor. Some restaurants might do it without egg as well but it is difficult. This is extremely delicious and there are some "make your own okonomiyaki" restaurants..

There could be some snacks such as rice crackers (senbe), Soyabeans (edamame) and hosts of salad bentos that can be useful plus nutritious for vegetarians.

Please note that the vegetarian term is confusing. "No meat" can mean Fish or egg or seafood is ok OR recently a coffee shop gave an adv of Veg Pizza - basically it had meat but also had vegetable toppings on it and hence they named it veg pizza. So to avoid confusion you can always say -
Niku nashi (no meat), sakana nashi (no fish), Tamago nashi (no egg) or
Niku nuki (remove meat), sakana nuki (remove fish), Tamago nuku (remove egg) ..
Many times on salads (especially caesar salad) there is bacon or raw egg on top of it. So pls ask them bacon nashi (no bacon) or tamago nashi (no egg).