Use a database

Best practice is to manage all of this information outside Altium using its database integration sub-system.

You can link symbols and footprints via a table rather than trying to manage this information inside of a symbol (which is the wrong layer of abstraction) or in a library of symbols (grouping becomes arbitrary and difficult to manage as the library size grows).

Doing it this way means that you never have to mess with internal attributes and double-entry of data.

My approach

I use CircuitHub for library management. It's free and it's trying to be a Wikipedia of parts.

It plugs into Altium (and other tools) and provides linked symbols and footprints via a local database that is sync'd from the web (so you can use it offline).

You can add/edit/manage parts on the website and it automatically (via Dropbox) syncs those changes to your computer where you can use them in Altium (or any other supported tool).

Here is how you setup CircuitHub to work in Altium.

Here are the basics of component management in CircuitHub.

Disclaimer

I helped create CircuitHub, because I got tired of dealing with these kinds of problems in library management.

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.