Can a foreigner living in the US with a student visa work as an independent contractor for a US company?

No. A student is not allowed to work outside the college. Even volunteering for a charity is illegal, if what the student does for free would be something that usually would be done by a person who gets paid for it. Otherwise people would work "for free," at least officially, and when ICE makes their rounds they would state: "I don't get any money here." Doesn't work