I am not set on making my logo or brand reflect this origin, but am open to ideas about how it could be incorporated. The vault tech logo from the game (attached, "VaultTecLogo.png") is a quasi 1950s emblem...if this could be updated and worked into the logo, that might be cool. I'm also very open to a super clean type only logo with some sort of unique flare.
As for the word "Tech", it should have minimal prominence. From a visual hierarchy, approach the word "Tech" as if it's "Inc" or "LLC" in another businesses logo.