The USA had a huge population, even in the 60's. We didn't need to mass draft for WW2, so why did we have to for vietnam? I have a hard time believing there were not enough volunteers.
We did, in fact, have a mass draft for WWII. It was our first in peacetime (1940, before we officially entered the war). 2/3 of the troops in Vietnam were volunteers, so the draft supplemented the regular troops.