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.