The bad news is that Vegetarian restaurants are rare in Japan. You'll often be forced to scan Japanese menus to pick out vegetarian items. Japan (being an island with few natural resources) is a fish eating culture. It's much easier to avoid meat than fish. A few Japanese sentences that may help:
niku wa haitte imasu ka (is there meat in this?)
sakana wa haitte imasu ka (is there fish in this?)
The Japanese word for vegetarian is "bejitarian". Be careful with this. Many people in Japan assume that vegetarians eat fish.
Shojin Ryori (Japanese Buddhist) food is always vegetarian (usually it's also vegan). There are Shojin Ryori restaurants in Japan (especially in historical cities such as Kyoto). It's also possible to eat at some Japanese temples.