Books similar to Wool, or the vaults in Fallout?

Try A Canticle for Leibowitz

City of Ember comes to mind

So bit of a stretch recommendation but I can never turn down a chance to recommend this author.

If you like the general idea of an author exploring what-if premise of people living (like how the Fallout vaults are given specific scenarios to live in), you might like Ted Chiang's short stories. For example, "Hell is the Absence of God" explores what life might be like if angels (heaven and hell) were real but come into societies like tornadoes or hurricanes due to their supernatural power (think tornado chasers). There's world building in it to realistically support the premise of the story (e.g. support groups for those affected) and is generally really thoughtful.

Again, very much a stretch to Fallout vaults, but really deserving of a read if you are more interested into exploring the realism in a world built around certain premises.

I've heard the Silo books go a similar direction. I've not yet read them; they're on my short list though

Tangentially related, but you might enjoy Margaret Atwood's Year of the Flood trilogy.

Lucifer's Hammer is very good and somewhat related.