H-1B holders can be passive investors in companies. That means they may not actively derive an income from such entities in exchange for working there, but they could be stakeholders or owners (e.g. a startup that they finance, but do not work for).

So, legally speaking you *could* declare a sole proprietorship and apply with your SSN. There are examples of H-1B holders who are getting approved for a business credit card. The main concern here is whether you would receive any tax forms from your account provider that you will have to declare, even if you receive zero income from your sole proprietorship. This may get complicated if you need to submit any tax forms you apply for a Greencard.

Does anyone have any details on this? Might be worth consulting a lawyer.