U.S. law is very strict about what you can and cannot do while in the U.S. on any particular type of visa. Generally, on an F-1 visa you cannot work at an outside job, so the legal hurdles you might encounter would depend on whether you both planned to have active roles in the company (prohibited) or merely serve as passive partners (possibly permitted).

One exception is that F-1 students on OPT (optional practical training) may be able to become self-employed business owners in limited circumstances. The student must show that he has the required business licenses, and actually does business in a field directly related to his studies. It’s quite possible that, as you don’t need a specific license to open a web company, your concept may not qualify as the kind of business that you may form as a student. The Immigration and Customs Enforcement Policy Guidance (2010) provides some direction about ICE’s take on this issue.

The penalties for getting this wrong include deportation, so if you’re really serious about this, run -- don’t walk -- to your nearest immigration attorney. There may be other visa classifications more suitable for what you want to accomplish, and depending on how much you want to invest (e.g., $1,000,000 for EB-5 visas). Regulations change frequently, and you need the most up-to-date information.