While we are not told specifically when the events take place, we are given one specific reference point and many context clues that would place the story in the late 1960s or early 1970s.
The narrator provides a specific reference point when he mentions Westmoreland's decision to dig in at Khe Sanh, a 1967 battle in Vietnam in which digging in was very probably a tactical error. This means the story had to have taken place in 1967 or later, and probably not that much long after, as otherwise the narrator most likely wouldn't have had that battle on his mind.
The whole late-night scene of sex and drugs and rock and roll, the desire to "drink beer, smoke pot ... [and] savor the incongruous full-throated roar of rock and roll" speaks of a late 1960s ethos. The price of a hamburger, 40 cents, also fits the late sixties time frame.