Most traditional Japanese supermarkets will have everything you need for cooking and sustaining a healthy lifestyle, but depending on your location, it is sometimes hard to find healthier, specialty products.
For vegetarian/vegan products, my current recommendation is Natural House. Natural House is a health store chain with a large selection of organic produce and food products, vegan/vegetarian packaged foods, bread, snacks, desserts, pre-made meals. The location nearest to me has a dining area where you can sit and eat. Not everything sold is vegetarian or vegan.
Most stores seem to be located in Kanto, but there are some locations in Kansai as well. http://www.naturalhouse.co.jp/shops/
Besides Natural House, you can also find numerous independently-owned stores scattered in Japan. Use https://www.happycow.net/ to search for locations in your area.
Sometimes smaller chain supermarkets will also sell specialty vegetarian and vegan items. Ohsawa Japan is a macrobiotic food manufacturer that sells a lot of vegetarian and vegan-friendly pre-made foods, like the ramen below, soup stock, pre-made curries, and more.
In the worst-case scenario, you can always buy specialty products online. Amazon Japan has a wide selection of items and ships very quickly. iHerb.com is also quick and reasonably priced and sells many vegan specialty items that can be shipped to Japan.
How to buy organic in Japan
Look for a green JAS mark when you go shopping. Organic products and farms are certified and inspected by an entity known as the Japan Agricultural Standard, which is, from what I can tell, compatible with other countries’ standards. While the organic market in Japan is small, it is growing. In many stores, you can find readily organic Japanese groceries, including soymilk, tofu, and rice. A wider selection appears at slightly more expensive stores, as well as bigger chains. Below is a list of some of the chains I like:
Ito Yokado: http://www.itoyokado.co.jp/store/index.html
Supermarket Kohyo: https://www.kohyo.co.jp/stores/
Aeon MaxValu: http://www.maxvalu.co.jp/
Seijo Ishii: http://www.seijoishii.co.jp/shop/ (English available!)
Shopping local and asking your neighbors
Another smart option is to keep your shopping local, which is actually pretty straightforward in Japan. There are a number of markets run by local agriculture groups. Look for the “JA Chokuhanbaisho” to see if there is one in your area. If you live in a rural area, get to know your neighbors! One day you might receive fresh ingredients and instructions on how to make it into something delicious.