Yes - in Arizona, Arpaio is already doing it. It doesn't even have to be at a "time of crisis"; Arpaio formed up his posse (under that name) before 9/11.
Sheriffs most anywhere have broad authority to hire and fire deputies. Here in North Carolina, training requirements for unpaid volunteer deputies would limit things somewhat as to calling them "deputies" - but volunteer squads along the line of the local Burlington police force's "reserve" (as opposed to "auxilliary") division are obviously possible.
As for "time of crisis," any sheriff in southern Arizona could easily say that illegals are overloading the jail he must manage - plus overloading local schools.
