Lucy and her family and friends are likely from Vault 4
The opening shots feature a decidedly un-mutated Walton Goggins delivering a monologue, seemingly on behalf of Vault-Tec, about the value of their products. It seems pretty clear from this shot and a later one that before the bombs dropped, Goggins’ character (then probably known as Cooper Howard) made a living as an actor and worked with the company responsible for all the vaults our dwellers emerge from.
It’s also likely that this was a specialized ad for the residents in and around The Ghoul’s area, which later appears to be confirmed as L.A. via a flashback. All signs point to Vault 4, as you can see from the door, being the closest vault, and the one where our story will pick up with Lucy some 200 years later.
