To reach the Key Vault, you must reach an alternate exit in the adjoined level.
So how do you reach the alternate exit? You need to take the blue, '?' marked path when the route splits.
The catch is that you may need to take more than one, and if you miss a jump or one of the jump pads, you may fall back onto the main track, rather than get sent immediately back to your checkpoint.
The alternate exit looks like this:
Once you return to the map, the way to the key vault will be clear.
The Key Vault is a level onto itself, and once you beat it, you will get a notification that you've unlocked the keys for that world. If you go back to the levels with keys, you will now be able to collect them in order to unlock the alternate paths.