In the United States, fallout shelter development came in basically two varieties. One was for regular people. Most of these were something that people were supposed to build themselves, in their basements or in special structures, on their own property. This was the bulk of the fallout shelter program in the early 1960s, encouraging people to build their own shelters, and took the form of giving out information as to how to do this (pamphlets, books) and I believe some tax rebates for funds spent on this. It got a lot of attention but did not end up with a lot of shelters built.

Under the same early 1960s Civil Defense program, existing spaces that could be used as short-term fallout shelters were identified in buildings. These are still marked today with the classic "fallout shelter" signs you'll see on many older buildings, especially municipal buildings, in American cities. These spaces were meant to be stocked with supplies; they are no longer.

In both of these cases, you're talking about shelters that were meant to reduce the amount of radiation received by people within them during the period in which the amounts of fallout radiation could be an acute danger (enough to make you sick or kill you). At most this lasts a few weeks. So it is not like the vaults in Fallout — they are not meant to last for centuries or be self-sufficient or anything like that. They are just places that one could stay in for a few days or weeks, with food and water and bedding, until it was safe to leave the area. (The area might still be contaminated with fallout, but the radiation will have dropped to levels that people can move through the area to less-contaminated areas, and decontamination can begin.)

The more Fallout-like approaches were not Civil Defense, in that they weren't for regular people. They were part of Continuity of Government, which is basically a catch-all phrase for policies meant to keep the US government alive and kicking after nuclear war, so it could coordinate a rebuilding of the nation. These include very fancy and specialized facilities, like Raven Rock, Cheyenne Mountain, Mount Weather, and Greenbrier, which were underground (many under-mountain) facilities meant to allow the core military brass, the President, Congress, cabinet agencies, and so on, to survive nuclear war, coordinate responses during a nuclear war, and coordinate the rebuilding of the country after the war. It is unclear to me exactly how long you were supposed to be able to live in these things, but presumably for longer than a standard fallout shelter — they were meant to be new seats of government, essentially, after DC was presumed destroyed. Again, the intensity of radiation from fallout drops off dramatically after a few days and especially weeks, so it is not like in the Fallout games (or the popular imagination) where you would be spending decades or centuries in there. (Raven Rock, the real facility, is clearly the inspiration for Raven Rock, the Vault in Fallout 3, but I suspect the similarity ends at the name and basic role.)

Separately, some states also have emergency management facilities that were built in this period and are also underground. I have toured the Massachusetts Emergency Management Agency (MEMA) headquarters, which was built in the early 1960s and meant to be able to survive fallout from an attack on Boston. It is entirely underground and a very large facility that includes all sorts of communication technology, bunks, even a small morgue. It was built so that the Massachusetts Civil Defense Agency would be able to survive a nuclear attack for a few weeks, and then be able to coordinate the state-level responses. Under President Kennedy, this sort of facility was meant to become common throughout the country, but he was assassinated shortly after it was completed (they have a letter on the wall from him talking about how great the facility is, and it's dated a few days before he went to Dallas, if I recall), and after Kennedy the enthusiasm for this kind of thing scaled back a lot.

Lastly, one other possible Fallout inspiration — there was, in the late 1950s and early 1960s, some interest in building underground facilities for general civilian use in the US. My favorite of these is the Abo Elementary School in Artesia, New Mexico, which was built entirely underground. It was in use until 1995, and there were many studies of it (and the handful of other underground schools built) to see if having a school entirely underground (no windows, etc.) impacted the psychological well-being of students and teachers (my recollection is that it isn't much worse than a regular school in that respect, but I haven't looked at these in a while). So one sees echoes of this sort of thing in the Vault school of Fallout 3, for example.

So, in short, one can say that the Fallout series definitely drew upon inspiration from real-life fallout shelters and bunkers that were developed in the US in the early 1960s, but nothing quite like a Fallout Vault — a long-term, self-sufficient, multi-generational underground community — was ever built or really contemplated. (I am leaving out a lot of other more science-fiction aspects of the Fallout Vault system as well, of course — deliberate experiments, etc.) One reason for that is that you don't really need that sort of thing. The Fallout games do not have a realistic depiction of the aftermath of nuclear war, or of radiation. The other reason is that building a Fallout Vault-like facility would require far more resources than anyone would be willing to spend, and technological breakthroughs that arguably we can't really even do today (they verge on the kinds of tech we'd need to have a colony on another planet, and we're not quite there — look into the problems that Biosphere 2 had, if you want some indication of the difficulty of running truly close-loop ecosystems).

I have not touched on the Soviet approaches here (you can find other answers on Soviet Civil Defense if you use the search function), but there are some similarities, and some differences, but they did not build self-sufficient "Vaults" either, for the same reasons.

A very nice history of fallout shelters in the US is Kenneth D. Rose, One Nation Underground: The Fallout Shelter in American Culture (New York University Press, 2001). For more on continuity of government plans in the US, check out Garrett M. Graff, Raven Rock: The Story of the U.S. Government's Secret Plan to Save Itself — While the Rest of Us Die (Simon & Schuster, 2017). On the Soviet approaches, see Edward Geist, Armageddon Insurance: Civil Defense in the United States and the Soviet Union, 1945-1991 (University of North Carolina Press, 2019).