The "Big Three" in veggie burger making are tofu, beans and mushrooms. Japan prides itself on tofu — and by extension beans — and mushroom varieties are a mainstay of the nation's cuisine. So it shouldn't come as too much of a surprise that Japan cooks up a variety of veggie burgers.
A quick internet search can reveal a fair number of options in Tokyo. DevaDeva Cafe in Kichijoji (devadevacafe.com) apparently does a mean vegan burger, and Island Veggie in Hiroo (islandveggie.com) has an extensive list of vegetarian burgers on its menu.
Burger-craving moments often arise unplanned and can happen in any neighborhood, so luckily two large chain restaurants offer a veggie option.