Vegans and those missing food from home will be particulary glad to hear that there are some places which will deliver all kinds of food, drinks and other products to your door.
The number one life saver which delivers organic vegetarian food and other things anywhere in Japan is

Tengu Natural Foods

There is also

Warabe Mura Wholefoods
The Foreign Buyers' Club
Indojin Indian Food
Baticrom South-East Asian/Northern African

See the Links section for more.

SHOPPING STREETS
Explore the old-fashioned shopping streets crowded with elderly ladies if you're lucky enough to have one near you. Here you'll find fresher and sometimes cheaper food than supermarkets. Look for fresh tofu shops, fruit and vegetable markets, and pickle, rice, and miso shops.

SUPERMARKETS
You'll find heaps of vegetarian raw ingredients for cooking. As for pre-made food, sauces, instant food etc, you'll have to read the ingredients because some things have surprising contents. If you cannot read any Japanese, your first trip to a supermarket will be rather confusing.

HEALTH FOOD STORES
If you're lucky, you might find a health food or organic food shop. As well as organic fruit and vegetables, you'll find dry foods, non-GMO tofu, bread, soaps and other things good for vegans and vegetarians. There is often suitable ramen, other instant foods and soy/gluten meat. You'll also find Sanitarium products (called 'San-Iku' here). Here is a list (in Japanese) of stores in Japan stocking San-Iku products. Natural House is a health food store chain.

INTERNATIONAL SUPERMARKETS AND FOOD SHOPS
Keep your eyes peeled for international supermarkets like Kaldi, Kinokuniya and Meidi-ya, and big liquor stores like Yamaya, as you'll find lots of international foods good for vegetarians plus lots of things you might be missing from home. If you are in an area with immigrants from Asian countries, look out for Asian grocery stores which also have plenty of good things for vegetarians. 

CONVENIENCE STORES
If you want something small to eat, then you can pop in at one of those ubiquitous convenience stores. They usually have a range of bentos and snack foods. For vegetarians though, it's limited to onigiri (umeboshi, konbu, sometimes there are other seaweed/pickle varieties), sometimes sandwiches, salads, pickles and bread products. Check the ingredients though. Of course there are also lots of snack foods and desserts which are ok.

SUPERMARKETS
Supermarkets usually have some ready-to-eat food. You'll probably be able to find onigiri, sushi, tempura, salads, sandwiches, pre-cooked rice, pastry and bread products, fresh fruit, fried food and other items. Just wander around checking the ingredients. Supermarkets often even provide a microwave to heat things up.

DEPARTMENT STORES
If you're near a major train station, head to the basement of a large department store such as Takashimaya, Mitsukoshi, Matsuya, Isetan, Tokyu etc. There will be lots of tiny food shops perfect for grabbing a ready-made meal. Look for the bakeries, onigiri, salad, tempura and tofu shops in particular.

BAKERIES
For a bigger range of better quality bread and pastry products which taste much nicer and have less artificial and animal ingredients, avoid the convenience stores and supermarkets, and head straight to one of the small or European-style bakeries. Of course plain bread and sweet foods can be found, but also savoury bread products and sometimes sandwiches. A knowledge of katakana will help avoid unwanted ham surprises. Japanese style bread tends to be very soft, a bit sweet and almost always white. Japanese bread products sometimes contain shortening which could be made from animal fat.

BENTO SHOPS
Bentos are ready-cooked or made-to-order meals available from bento shops like Hokka Hokka Tei, Origin Bento, and independent shops. Usually all bentos are not suitable for vegetarians, but these shops also often sell onigiri, salads, tofu and self-serve dishes which are sometimes ok for vegetarians.

SUSHI SHOPS OR DANGO SHOPS
You can get inari, kappa-maki, kanpiyo-maki, natto-maki etc from small take-out sushi shops. Dango shops are small shops which sell Japanese sweets made from rice and beans. Often there is a small take-away counter which sells sweets as well as some kinds of sushi.