Student with online business

I will begin studying aboard in Japan starting this April. I currently run an online business while attending school. When I move to Japan will I be able to run the business with no issues? I require no room for inventory and such and all the money goes into my US bank account.

Why should it be a problem?
Just make sure you pay your taxes in the US properly.

Immigration rules as they are have not taken remote work/online business into consideration at all, so the rules around them are not clear-cut. But since you will be on Student status, doing part-time work would be acceptable on that status, so I see no issue as long as you maintain good attendance and good grades :). Note that if you continue living in Japan for 5 or more years, then (but only then) you will need to declare all worldwide income to the Japanese tax authorities and pay taxes. But until then, as the earlier poster says, you will be paying income tax on that income to the US.