In some inner city neighbourhoods around Tokyo or Kyoto you may not see any supermarkets but you may see old style stores that specialise in one type of product eg. rice or tofu, fresh fish, or fruit & vegies. But there will be a supermarket somewhere if you really want one.
However, having said that, it's Convenience Stores that you will find the most useful (for a short term stay, unless you want to do a lot of your own cooking from scratch.)They are everywhere and they'll have everything you need to survive - you'll be able to put together a gourmet meal every day.
Most convenience stores have a good range of fresh, ready-made meals and salads, you can even choose the dressing you prefer. There are ready made Japanese meals & Japanese versions of Korean, Chinese & western dishes, eg. spaghetti, fried chicken. Also sandwiches, rice balls, curry rice, side dishes -pickles, kimchi, various soups, fresh bread, cakes, yummy desserts, fruit, condiments, butter, cheese, ham, eggs, tofu, softdrinks, alcohol -beer, wine, umeshu. Look, you can live out of a convenience store and not have to ever eat anywhere else.
My first introduction to Tokyo over 20 years ago was Winter Oden stewing on the counter at the 7-11, My thoughts.. What the..?! It smelled like someone was stewing their old socks. But I was soon addicted to late night Oden. When the season changed they had Chinese style steamed buns on the counter.
Have a look at youtube there are a few uploads & see how great Japan's convenience store food is. (Notice I did not once mention anything about this stuff being healthy but it's tasty)