I've had my fair share of veggie food in my travels to Japan, including some good old traditional treats:
various sobas
tempura
vegetable sushi
all the rice you can eat
veggie yakisoba
udon can be a little tough with no meat or fish broth
curry rice
pickled vegetables
yummy Japanese strawberries and Asian pears!
THE BAKERIES
Sorry tigermilk, I do not mean to negate your experience in Japan, but some of the food you listed is not common or not easily available...
Normally tempura includes seafood, and "vegetable sushi" is not common - if you go to sushi restaurants in Japan and if you want sushi without seafood, then what I can think of is
kappa maki (sushi roll with cucumber) or kanpyo maki (another roll with stewed kanpyo which is thin strips of dried gourd).
I've never heard of vegetable yakisoba, usually they put some pork.
You'll find meat in most curry rice (sometimes they serve "vegetable curry" though), and if udon is off because of fish based broth, then same goes with soba.
If you can eat seafood, it is much easier.
Also, if you're vegetarian and like tofu, and want to experience traditional Japanese food, I'd recommend to visit restaurants specialized tofu dishes. They serve tofu and yuba in many different ways.
http://www.umenohana.co.jp/e/index.html