First, be advised the United States is the Federal government -- not the several states. Who does the calling into actual service? Certainly a state does, but this does not preclude the state's right to call their own militia to settle their own border problems first. In that the state militias, later named 'National Guards' are already federalized, it remains for the Citizens of the state to form their own protective services, 'necessary to the security of a free state,' (Second Amendment, not controlled by the Federal government).
If the feral gummint refuses to act, it can only be because the governor refuses to call on the federal government for assistance, notwithstanding the presence of the border patrol. If the governor refuses to activate the state's defense forces in such a crisis imposed upon us by Mexico, by default, it's the Citizens who are responsible for their own safety and protection. That right and responsiblity falls to the people when all other agencies fail. No permission needed, thank you.
The sheriff should be aware of this right and should act accordingly. If the sheriff refuses to start deputizing and managing the problem, re-call the sheriff. He was elected directly by the people and first responsibility is to the people. At no time in our history has a Citizen been prevented from exercising the right to defend lives and property against any threat, apart from the war between the federal government and the states and the advent of gun control laws in some some states.
Hey, if there is no people left when this is all over, there will be no need for a government of the people, by the people and for the people. And some would already agree we don't have a government of the people, by the people and for the people because we wouldn't be pulling our hair out over this illegal entry and open borders situation in times of great peril if we did.
Just another opinion, but hopefully, based on the principles of self-government when there is no government assistance when it is required. I would be forced to say to the feds, the state, the county sheriff and the supervisors, if you are not going to lead, get the hell out of the way.