National Guard units can be Federalized through a formal and limited process - i.e. brought under direct Federal command - but this is rarely done (e.g. Vietnam, and a few went to Iraq).
While remaining under the command of the State Governor (the usual state), they are exempt from Posse Comitatus, and can enforce laws domestically.
When operating under their governor, they are paid by their governor. That’s why such a deployment would beg to be federally activated. DOD has a much bigger pot of money.