Alright so I went back through my DSM-V and I stand corrected- Vox absolutely has NPD (Narcissistic Personality Disorder).
Now obviously I'm not a psychologist so I can't give out diagnosis' much less to fictional characters who I am not the doctor of-) But I have a dsm-v copy, have been studying psychology since I was a small child, and have been analysing characters for basically just as long. So I can say that he meets almost every single criteria for NPD except for (imo) criteria number 7.
Was looking for other things I thought he might have and I think he MAY also have OCPD (Obsessive-Compulsive Personality Disorder), but I don't think he meets criterias 5 & 7 and I am unsure on criterias 2 & 4.
And I still think he has BPD, he meets most of those criteria as well with the exception of maybe criteria 3, maybe 9, and MAYBE 7, but I'd wager a hard bet that he does meet that one even if we don't necessarily see/hear such feelings expressed. Throw in the fact that Alastor ABSOLUTELY fits being a favourite person based on how Vox acts due to/in regards to him? Hundred 100% he has BPD. This one, while again I can't diagnose anyone (much less a fictional character), I can at least "peer review" as someone who does have BPD.