Managed to grab a key for SWORN this month (1st success). Seems your best bet to get something from the Game Vault is: 1.
Decide what you want ahead of time. 2.
Figure out when it unlocks in your time zone. 3.
Make sure you access the page at the opening minute. 4.
Wait out loading of the page rather than refreshing.
The site will be overloaded by everybody accessing it at the same time. 5.
While waiting for the page to load...pray that there's still a key left by the time the page actually loads...