The Vault allows you to store Food, Gold, or Oil to hide from enemies. In other words, resources stored in the vault cannot be looted by the enemy under any circumstances.
You can only store one resource in the vault at a time.
You have to wait one day to withdraw the resources that you put into the Vault (Unless you pay an early withdraw fee in Crowns).
There are 2 ways to store Oil in the Vault: have a Vault level 3 or higher, or research Savings chapter 3 at the Library.
Interest gained is stored back into the vault thus increasing the vault's official capacity. In other words, storing 6000 oil becomes 6300 at 5% interest. You must withdraw to gain the interest as interest is only added once (after 24 hours).
Museum artifacts that add interest only need to be enabled when the 24 hour clock ends; so enabled exactly 24 hours after you store your resources.
Note: there is little reason to upgrade the vault beyond level 1 as storing oil is its main purpose; and upgrades do not increase oil storage capacity. Simply research all 5 Savings Library chapters to store 8000+ oil.
Chapter 1: Vault generates 5% interest after 24 hours.
Chapter 2: Increases Vault capacity by 100,000.
Chapter 3: Vault can store 6,000 .
Chapter 4: Increases Vault capacity by 100,000.
Chapter 5: Increases Vault capacity by 2,000.
Emperor Moctezuma at the University confers these bonus:
Vault Capacity: Increases Food, Gold, and Oil storage capacity in the Vault by 70%(10 levels). Only applies to the base oil storage capacity (not the 6,000 then additional 2,000 storage bonus acquired in the Library).
The Eiffel Tower wonder also adds 200,000 and storage to the Vault.