Very unlikely. Last update given to this mod was 8 years ago.

I'm looking for a mod like this for my vBulletin 4.2.3 site. So far, I've only come across one which allows users to change their username themselves, rather than admin approval in the AdminCP.

King Kovifor (Jeremy) is now a moderator at Xenforo
https://xenforo.com/community/thread...kovifor.56372/
Having moved on several years ago, and having not been online here since August 2015, I would guess he is unlikely to be doing any more vBulletin development.
These addons may be an option
Username change requests without moderation
Username change requests with moderation

The links you provided aren't for vBulletin?

Unfortunately no. I believe that vB is a dead product and that like we did in 2013, most everyone else is going to have to migrate away from it at some point. I'm showing you that what you want to do is possible, but with different software. (Although it's from the same guys that originally brought you vB3).

I recommend posting an unpaid request, and state exactly what features you want.

You never know, one of the coders here might take on the project for the benefit of the community. If not, you could then post a paid request. :)