Leaks from the Fallout TV series (not pictured because sites which have been linking them have been nuked by Amazon) have seemingly confirmed that the show will be partially set in and around Vault 32. This is a brand-new vault, not seen in any prior Fallout video game. This suggests the show will be pursuing a new story and location rather than adapting a game storyline directly. Since every instalment of the Fallout franchise changes location, characters and storylines, that's a reasonable road for the TV show to take.
In the Fallout universe, much of the world is laid waste in the Great War of 2077. Tens to hundreds of thousands of people survive in 130 large underground "vaults," located below the United States of America. Over the course of the next 200 years, people emerge from the vaults at different time periods and begin resettling the surface, but have to content with other survivors and radiation-created monsters. Only 64 of the vaults have had their locations confirmed in the games so far.
Following the vague numbering system used in the games, which generally move from west to east, Vault 32 should be located in the western part of the former United States. Vault 34 is located in Nevada and Vault 29 in Colorado. We know that the Fallout TV show is partially filming in Utah, and Utah would make a splendid location for a vault. Vault 70 is also located in Utah, albeit in background materials of dubious canonicity.
The TV show is also filming in New York and New Jersey, leading to speculation that the Fallout show may revolve around a road trip across the American wasteland. This would be a marked shift from the games, which are set in and around a very specific location.
The Fallout TV series stars Walton Goggins, Ella Purnell, Kyle MacLachlan, Xelia Mendes-Jones and Aaron Moten in undisclosed roles, with more cast to be announced.