I did a similar thing in Korea. The legal system is very similar. Korea inherited a lot of their modern legal system and bureaucracy during the military occupation during the first half of the 1900s.
In Korea, here are the traditional steps. AFAIK they are REALLY similar in Japan.
Get an office. You can probably use a home, but someone from the tax office is going to come inspect once a year to make sure that it really is a place of business. (Depending on what you do, it will matter what the office looks like. If you claim to be manufacturing chemicals, but you’re doing it from an apartment address, chances are you’re not going to get approved.)
Register your corporation with the Division of Corporations (In Korea, the 등기국). There’s quite a bit of paperwork, but that’s the case with everything in Korea and Japan. They have a counseling office where they can help you. But chances are, you’re going to need to speak the language. (Seriously. I discussed this with EVERY person along the way in Korea. Everyone said, “Oh, thank heavens you speak fluent Korean. We get foreigners who wander in every once in a while and it’s impossible. It takes FOREVER.”)
Receive a license from the district office. This is not the “business license”, but a license permitting a certain type of activity. This only matters if you’re trying to get something like an education/cram-school license or a food handler’s license for your restaurant. Or a liquor license for your restaurant. (Or a license to manufacture chemicals in your home, I guess.)
Register for a business license with the Tax Office. You’ll need the business registration certificates that you received from the Division of Corporations. This is where you are actually granted a business license.
Congratulations. You’re a registered business in Korea. Or hopefully, Japan.
Then you have to go through the business of applying for the visa. That requires all SORTS of documentation from your home jurisdiction, including a bunch of letters and forms that are apostilled. (It’s an internationally recognized notarization.)
In Korea, and likely in Japan, you can hire a legal barrister for it. (Somewhere between a paralegal and a lawyer.) In Korea, I’d plan on USD$4000, and plan on them saying “No, this totally won’t work” all along the way. Also, watch them be amazed when it works, because it actually does work.
Or, you can do like I did and do it all manually. By hand. Painfully. Visiting all of the government offices. And doing all of the translation of all of the documents by yourself.
Not a horrible experience. I learned a LOT. Including how to translate Articles of Incorporation into Korean.
But you probably won’t want to do that.
Best of luck, and please post what you learn here!