Now I know the core set caters to 3 players, but how many players can I add (for example) By adding the player's vault to my collection?

Thanks in advance

The Player's Vault, as I understand it, should have enough cards to cater for three players, like the Core Set.

So, if you have the Core Set, and you buy the Player's Vault, you will be able to comfortably cater for 6 players.

Or the core set or players vault + colour scanner/printer and you can have as many players as you like

Of course, it's nicer to have the actual cards, I suppose.