If by Vietnam War, you mean the entirety of the conflict between North and South Vietnam, then it is certainly the South Vietnamese troops. Since the insurgency began in the late 1950s, they were responsible for all the fighting and again when the 'Free World' forces left in the early 1970s.

The simplest indicator of this is that ARVN suffered about five to six times the number of casualties, not to mention the police forces, than that of the US armed forces. And of course there was a tremendous number of casualties among South Vietnamese civilians.

Though for a few years after 1965, US ground forces did push ARVN to the side strategically, and emphasized their role as 'pacification' forces -- in other words, the guys who go in after an area is taken from the enemy, build wells, provide medical services, and police the area to keep it secure.

In short, if you are limiting your question to 1965-1968, then you could argue the US. But, if you consider the entire war, not just the small period of direct US involvement, then it is clearly ARVN.

All of the airstrikes carried out in the North and most of those in the South were carried out by the USA, which between its Air Force, Navy, and Marine Corps had far more combat aircraft thn South Vietnam. On the ground the USA did a great deal of, if not the bulk of fighting until Nixon was elected in 1968 with his policy of "Vietnamization" of the conflict, after which the South Vietnamese army (ARVN) took on a far greater share of the fighting until the fall of Saigon.

Let's not forget the contributions of Australia and South Korea either. Though SK's commitment was only a tenth of that of the US in total troop numbers, as a percentage of their national population they made a huge sacrifice in manpower to back up the USA, especially considering that their chances of winning a war with North Korea then we're nowhere near as good as they are today.

While ARVN was a respectable force in terms of manpower and equipment, there was no escaping the fact that it was a less effective military force than the VC and NVA for the most part. Corruption was probably the most important factor in that but there were many other.