First thing about vibrato is there are no rules. Different players do it different ways & the players who are known for having great vibrato (Yngwie, Blackmore, Marty Friedman, Angus Young, BB King etc..) always have a distinctive vibrato of their own (different from each other's) that's often a big part of what makes them sound like they do. It's there to accentuate what your'e playing...ie make it more musical & expressive and to hopefully give it a more personal stamp.
I kinda find it hilarious when I hear of people trying to mentally count/time their vibrato like tap tempo of something. There's no better way to instantly sound wooden/clinical imo..but eh, to each their own. Unfortunately that's the sad state guitar playing has got to these days..you get everything else out a book..so why not vibrato? Well too bad, but it doesn't really work that way.
The best thing to do is to genuinely feel what your'e playing. Develop one's ear so it's sensitive to pitch & the notes being played 'cause good players aren't thinking about their vibrato (as they play) at all..it just happens naturally...
Don't know what styles you're into but jamming the blues is a great way to get your'e feel quotient up...vibrato, bends and all.