The President is, by the Constitution, the Commander in Chief. But you say that we were not supposed to have a standing army (or military for that matter). Then, how could he be Commander in Chief of a non-existent military? He is the Commander in peacetime, too, you know?
But to be short, because we have much larger fish to fry at this moment, do not make assumptions about me or anyone else that you cannot back up with proof. The Constitution provides for the common defense, and that's what's happening.
End of story.
By calling up the citizen militia (all adult men who own a weapon) the President is the commander of those armed forces. Question answered. Constitutionally.