On the other hand, you could make a plausible case that the Constitution forbids Congress to send conscripts off to fight an undeclared war, or to conduct a peacetime draft. The theory here is that while Article I, Section 8 of the Constitution empowers Congress to "provide for the common defense" and "raise and support armies," the framers simply meant to establish a professional army, composed of volunteers. This is not as off the wall as it sounds. You will recall that early Americans found the British Navy's practice of forcibly inducting men into service pretty obnoxious, and it is unlikely they would turn around and authorize a draft.
The Constitution does permit the separate states (or "the several states," as we legal scholars like to put it) to organize militias for home defense (note the Second Amendment), and in the broadest sense the militia in colonial times consisted of every male 18 and over who was healthy enough to carry a gun. So (in this view, anyway), while the federal government can't draft people, the states can.
What enables the federal government to conduct the draft is clause 15 of Article I, Section 8, which permits Congress to call out the militia to "execute the laws of the Union, suppress insurrections and repel invasion." This means that Congress can send draftees off to foreign countries after a declaration of war, which has the force of law. But it doesn't include undeclared conflicts like Vietnam. And presumably it wouldn't include a peacetime draft.