There are heaps of little cafes around as well as the omnipresent Starbucks, Doutour, Tully's, Excelsior, Veloce etc. They sometimes have sandwiches, muffins etc which are ok. Look for the more bohemian cafes or cafe/bars because they are more likely to have vegetarian options. A lot of Japanese cafes often serve light meals like spaghetti which are sometimes ok.

There are lots of great vegan, vegetarian and veg-friendly cafes in Tokyo so have a look at the recommended cafes listed on this site.

You'll find Johnathon's, Denny's, Royal Host, Skylark, Gusto, BLDY, various other chains and small Western-style cafe-restaurants everywhere. These are places to go for drinks and desserts mainly because food for vegetarians is fairly limited. Chips, salad, and pasta can usually be found. Saizeriya is an Italian-style family restaurant which offers a few more veg options than the others.