I looked briefly for the number issued of each, but if that information is anywhere it’s likely buried in one government report or another. I’ve found a bit of information but not anything that definitively states one or the other. The wording of several sources indicates that the M16A1 would have been more common. I would tend to agree given it was adopted in 1967, and prior to that the XM16E1 was not standard among all Army units from their first deployment, and the Marines were still using M14s as well.
An alternative rifle was available however. The 5.56mm XM16E1 “black rifle” replaced the M14 in airborne and air assault units in 1964. The first three Army units deployed to Vietnam in 1965, the 173rd AbnBde; 1st Bde, 101st AbnDiv; and 1st CavDiv (Ambl) were armed with the “black rifle.” It had been in use in Vietnam by advisers and Special Forces since 1962. The only divisions deploying with M14s were the 1st and 25th. By August 1966 all Army combat units in Vietnam were issued the XM16E1. In December it was recommended that the M16 replace the M14 in all Army units. The XM16E1 had teething problems, which were largely corrected, and in January 1967 the improved rifles were redesignated the M16A1…
…The M16A1 was a good rifle, but it required meticulous cleaning, was comparatively fragile, and had poor penetration through brush, bamboo, sandbags, and earth parapets. Most troops arriving in Vietnam had never seen an M16A1, having used the M14 through Basic Training and Infantry Advanced Individual (AIT) Training.
[Source: The US Army in the Vietnam War 1965-73 by Gordon L. Rottman]