How can I create an Integrated Library from a Vault component?
I did some more research on this. It seems like there is no easy way of converting a vault component into an Integrated Library. Easy way in the sense of for example right clicking a vault component and choose something like "Fetch current revision to local Integrated library".

There are ways within the Altium Vault window to find the schematic symbol and layout footprint. Look for the link within the preview section within each vault component. From there you can download the symbol and the footprint and put them together locally.

A better way (in my humble opinion) is to go to the Altium Design Content page. Here you will find all Altium Vault components available for download. Just find the library which contain the component you are looking for and download it. You will get a .zip file containing an .intlib file which can be extracted to a .schlib and a .pcblib. From there you can copy both the symbol and footprint into your own libraries.