Tutoring US school students would be akin to working in the US and that would be illegal if you do not have a US work permit... So, I do not think that there would be any online tutoring company which will accept foreigners as tutors..
I'm bit skeptical about it and don't really understand how do you see it illegal as you'll not be working with US students while being in the US (so no question of work permit I think),rather it's more of a virtual classroom over internet...
P.S: I guess you're well aware of TOI in India...few months back I read somewhat like this over there.
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