In Fallout Shelter can you find other vaults at anytime in the game? And if so do you attack them as if you are a raider or what.
Quests involve a team of dwellers leaving your vault and visiting some pre-determined location to achieve some objectives and/or follow a story line. Many of the locations are other vaults, so yes, other vaults are found in Fallout Shelter.
As to what you do when you get there, it's mostly killing raiders and/or creatures that have invaded, while searching for loot and/or survivors.
In addition to OrangeDog's answer general exploration also offers the chance of discovering a Vault or dwelling to explore, similar to Quests.