They need to be Part 141 to be able to be SEVIS approved, which is needed for them to issue an I-20 to sponsor your visa.
For TSA approval, Part 61(not 91...) is enough. You can train in a P61 school that is not SEVIS approved, but this still requires a visa. The only way this is viable is if you start training in a Part 141 school and finish your training in a Part 61 school.