F1 international student to start a online business in USA
Regardless of your status, you can own shares or membership interest in any US company (as long as its not S-Corporation).
That means you can own an LLC or Corporation, as an example, and run your online business using this entity. Keep in mind though that your immigration status might not permit you to be employed by your own company, however it might permit you to draw dividends.
Bottom line is, I would recommend you to consult an immigration attorney and an accountant before drawing any profits from the business. However, there should be no problem with you owning the entity.