I'm with /u/platitood on this question. The draft was initiated in the lead up to WWII and b/c of US military occupation and peace keeping duties it wasn't immediately ended after the war. Then the cold war happened and the draft became a fairly permanent part of US culture. So, there actually wasn't a draft "for" the Vietnam war. There was just a draft that had been a US rite of passage for men since 1940. Vietnam created a need for more troops, but most of the draftees didn't go to Vietnam. Only about 25% of troops in Vietnam troops were draftees. B/c of the war's modern unpopularity a misconception has grown that it was initially unpopular. That wasn't true and many happily went.

One thing that complicates this is a lot of volunteers volunteered b/c they knew that they were likely to be drafted and they would have more control over where they went and what they did if they volunteered. My dad was one of these. He is Chicano and at the Black Americans (especially in the S. where there is a history of discrimination in who they drafted to place most of the burden on the Black community) and Latinos knew they were being disproportionately placed in the more dangerous zones to protect white lives. So, my dad, knowing he would be drafted, volunteered for the Navy so he could avoid being sent to the infantry. Some estimates are that as many as 55% of volunteers were people like my dad who knew they'd be drafted and the way to assert some control over the process was to volunteer.

There was good /r/askhistorians answer on this a while ago that's worth checking out. https://www.reddit.com/r/AskHistorians/comments/by8kqr/were_there_many_volunteer_soldiers_during_vietnam/