Button to Show Home Vault
 

Similar to the commands to open one's bank, is there a command to open one's vault?

Pretty sure no. Someone correct me if I'm wrong, but I believe the House Vault is simply using a container window.

Quib

The initial bug where it was recognized as the same window as the left-most inventory bag leads me to believe that's the case, Quib. (Plus it uses any installed bag mod when displaying.)