How might the election of 1968 have affected the course of the Vietnam War?

It is difficult to answer this, it would be easier to answer how did the affect the course of the 1968 election.
Nixon secured a narrow victory in 1968 over Hubert Humphrey and the Vietnam war was probably the most dominant issue in the USA at the time.
Although Nixon developed the policy of Vietnamization whereby the USA could withdraw over a period of time, leaving the South strong enough to withstand the communists, in ’68 he and his principal foreign policy adviser, Henry Kissinger, were firmly of the view that the war could be won.
Therefore Nixon’s victory in 68 did little to scale down the conflict. In fact if anything it intensified with the illegal and clandestine extension of the war into Cambodia. Carpet bombing, the CIA backed coup which put the Cambodian military in power and the invasion of 1970 led to the destruction of a fragile balance in Cambodia. This led to the devastation of the country and the coming to power of the Khmer Rouge.