I know what you mean.

As I see it, this is an area where a player adds their own flavor and is part of what defines their sound and tone. An instructional blues video I used years ago (Keith Wyatt I think) suggested loosening your wrist an using your wrist to get the vibrato.

I understand the concept, but that never really worked for me. What I find myself doing is lessening the pressure of my thumb on the back of the neck and using my arm and wrist to generate the vibrato. Sometimes I completely remove my thumb from the back of the neck when doing this. I find I can get a pretty fast vibrato doing that.

The one that gets me if non bended vibrato. I see players that spread their fingers apart, shake their wrist at some insane speed and get an insane vibrato. Joe Bonamassa comes to mind.

Hope some of that helps.