Were Americans drafted into the Vietnam war?

Yes. Initially there was a draft where all males of the appropriate age were eligible for the draft unless they were physically or mentally unfit, or had a deferment (formal excuse) such as being a college student. Later the Draft Board started a lottery, all were given numbers drawn from birth dates. Then they started to draft at #1, then 2, and so on until they had enough people to staff the military for the war effort.