Yes you can work the U.S. as a non-citizen if you have either a Green Card or EAD Card.
Some jobs require citizenship, for example, CBP Air Interdiction Agent or anything related to national security. But typical 91, 135, 121 shops won't have any issues with Green Cards.
Now I you don't have Green Card or an easy way to get one, like a U.S. citizen spouse or smth then your options are extremely limited and time consuming.