Usually you cannot plug in an external hard disk and play a game on another computer. This is because games often use DRM to restrict the number of devices (computers) you can install or play the game on.
In your case you mentioned you can authorize 3 computers for your game. Installing a copy on each computer is the way to go.
Unfortunately, it will take a little effort to resume gameplay from a different computer. You will need to copy the saved games files from one computer and save it to another before resuming play. There is no mechanism to allow you to save the games to an external disk instead of your local disk. Installing to the external disk does not change this either.
I don't have an experiences with Larian Vault. However, I'm sure a member of the Larian staff would be happy to assist you once they read your post. Or you can try emailing them.
