People who held communist beliefs or who belonged to or were associated with communist organizations were not exempted from the draft for these reasons. For example,
In July 1966, three working class draftees, James Johnson, an African-American, Dennis Mora, and David Samas refused to serve in Vietnam and were courtmartialed. Mora was a member of the DuBois Clubs, and Johnson subsequently served as an editor of the Daily World. Their case became a national cause of draft resistance and opposition to the war within the military, although all were court-martialed and given long sentences to be served at hard labor.
Source: Brian Rubinsky, 'Working Class Internationalism: The American Communist Party and Anti-Vietnam War Activism 1961-1971' (MA dissertation, Rutgers, 2014)
The DuBois Clubs "was a national youth organization sponsored by the Communist Party USA (CPUSA)" the The Daily World was a communist newspaper (known as the Daily Worker until 1968) which merged with the People's Daily World in 1986 to become the People's World.
Dennis Mora, David Samas and James Johnson. Source: displacedfilms.com
Known as the Fort Hood Three, Johnson, Mora and Samas actually denied being members of the Communist Party, but for a reason:
If these young men were to participate in such a massive anti-war crusade, alongside with other American radicals, denying Communist Party membership was imperative. Dennis Mora stated that “he was not a member of the Communist Party, and Private Johnson and Private Samas said that they were not members of any organizations other than the Army.” For these young American Communists, avoiding Party membership was the only way that they would be able to participate fully in the anti-war movement.
However, those with religious or moral beliefs against active combat could be excused combat service, though they had to prove these beliefs were sincere. Also, this could not be used to avoid being drafted; instead, such individuals might be assigned a non-military role (such as cook).
According to data presented in Michael S. Foley in Confronting the War Machine: Draft Resistance During the Vietnam War, almost 25% of 117 draft resisters surveyed in Boston identified themselves as socialists or communists (46% saw themselves as Democrats, 14.5% self-identified as anarchists and less than 2% as Republicans). Apparently, though, very few of the 25% socialists / communists saw themselves as committed communists. Foley observes that:
...it is apparent that this did not mean that these New Leftists were hell-bent on revolution. Actually, only about 20 percent of them felt comfortable with radical labels like ‘‘revolutionary’’ or ‘‘politico.’’ Rather, more than two-thirds of Boston’s draft resisters saw themselves as ‘‘activists.’’
I have not found any communist party members who actually served in Vietnam. Although lack of evidence is not, of course, evidence in itself, it is highly probable that they either resisted the draft or found a way to avoid it.