The Vault is where all of your collections are stored, whether or not they have been completed yet. Cards in the Vault remain there permanently; once they are moved from the Pack to the Vault, they cannot be returned. Cards here cannot be bought, sold, auctioned, traded, or stolen.
If you go into your Vault, the default screen will show active collections; if you click on one of those collections, it will show you which cards you've vaulted for it, and which ones you're missing, but it will not show the different varieties of the same individual card from the collection view.
To see a given card's different variations, click the dropdown menu that says 'Active Collections,' and then select 'Bling Cards.'
Clicking on Bling Cards will show each card that has distinct bling. So if you vaulted 15 of the same card, but used a different bling on each card, then selecting "bling cards" should display all 15 of those cards (and any other blinged card from any other collection), but it won't show these bling variations from the vault view of a specific collection.