Well in the Fallout Universe https://www.youtube.com/watch?v=Kq24DB2X760at least according to Fallout Tactic's introduction. So if we take that at face value, the War in Vietnam turned out differently than it did in our universe. America taking the Vietnam war far more seriously than they did in our universe, I can see America actually clearly winning the conflict early on. Thus no "counter culture" formed an thus no anti-Vietnam songs.
Lonesome Road seemed to hint at hippies:
Hidden Valley also had hippy graffiti. That being said, much as I like them, I don't think songs like Scarborough Fair or For What its Worth would sound right it in Fallout. Part of the humor of the songs is the juxtaposition of horror and Leave it to Beaver-esque camp.
Hippies were mentioned by NPCs in Megaton, who often tell you to "not let them fool you with their hippie crap", so some form of counter-culture did exist.
But it wouldn't have been the same counter culture that happened in our timeline and thus the songs that came out of our counter culture wouldn't exist in the Fallout Universe.
The Fallout Counter Culture might have been something that formed much later in the Fallout Universe or it didn't take off like it did in ours and just stayed a fringe culture that grew as the Sino-American War expanded.
Fallout's 4's intro tells us that by around that time "people woke up from the American dream." Realizing that they were running out of resources and so on. Fertile ground for a counter culture to grow from. The ideas of peace and love, don't trust the government and use less resources and all that hippie stuff would take root with a world falling apart.
I still like to think that America actually went to congress and declared war on Vietnam like the intro of Fallout Tactics says and won it early on.
The way I understand it, the only reason we only have very old 40's-50's songs in Fallout is because those songs were on old 78 speed records, which had fallen out of use long before the war and were stored in deep forgotten basemants, safe from the bombs. As such, while there were almost certainly many thousands of songs recorded in later years, they were lost in the fires and EMP waves.