That isn't the same as the rule of law. I have shot at (well near) people I thought were looters. Maybe they were and maybe they were not. The "lawfulness" of my act was borderline.
There could be some rough spots. Restoration of the legal system would come later. There is no way the officers of the state can cover the territory in that kind of crisis, and they actually cannot do it now without the aid of the ordinary citizens.