As I’ve stated many times here over the years, the basis of most Japanese savory foods is a good dashi, or stock. Dashi is not just used for soups, it’s used for stewing, in sauces, batters, and many, many other things.
The regular way to make dashi was one of my first entries  on Just Hungry. It uses kombu seaweed and dried bonito flakes (katsuobushi). Some people use niboshi, small dried fish, in addition to or instead of bonito flakes.
Katsuobushi and niboshi are both fish-based, so not vegetarian. So how do you make a good vegetarian, even vegan, dashi?
My preference is for a simple kombu based stock, which can be made quite easily by soaking a piece of good quality kombu in water in a closed container overnight in the fridge, as shown in the photo. (See this post  for more about kombu.) I put about a 30cm / 12 inch long piece of kombu in about 2 litres (8 cups) of filtered water. This is more kombu than for the regular kombu and katsuobushi stock, since the kombu has to produce all the umami on its own. If you don’t have time to soak it overnight, you can soak it for a minimum of half an hour at room temperature.
To use, bring the dashi up to a simmer (not a full boil), then take out the kombu. This produces a light colored, subtly flavored dashi. The soaked kombu can be cooked further until soft, and used in stewed dishes and the like.
If the kombu dashi is too subtle for you, you can try tossing in a couple of good quality dried (never fresh) shiitake mushrooms in the water. This needs to be soaked for at least a couple of hours, or until the shiitake are quite soft. This will produce a light brown colored dashi with an assertive shiitake flavor, which is especially suitable for dishes that will include the shiitake. The soaked shiitake can be used in your dish.
Finally, if you are using shiitake mushrooms for a dish and reconstituting them anyway, save the soaking liquid to use as dashi in the future. I keep the cut off hard stems in it to extract a bit more flavor. It will keep for a few days in the fridge, or can be frozen. (I freeze it in small ziplock bags, laid flat.)
If you cook whole dried soybeans , the cooking liquid is full of umami and makes an interesting rich vegan stock. Use on its own or in combination with one of the other stocks.