You do not need a school to sponsor your visa if you are a citizen of one of the 35 countries participating in Visa Waiver Program. It is possible to apply to the TSA for category 3 (below 12500lbs MTOW) training when you are already in the US.
https://www.flightschoolcandidates.gov/afsp2/
Click on FAQ and you should find answers to all your question.