This is one of the most difficult phrasing techniques to do well on guitar. Index finger vibrato on the first string requires moving the string only in the upward direction (otherwise you pull the string off the fretboard if you pull downward). It’s also difficult because it tears up your finger fairly quickly so you can’t practice it for hours on end. However, bit by bit my phrasing is getting better.
I usually practice for just 2 or 3 minutes at a time. I use simple minor blues licks and add vibrato on the root immediately after striking the note. I try get as much flesh as I can under the string, then push up on the string by turning my wrist counter clockwise (right handed players). Then I rotate my wrist slightly to release and apply pressure over and over again to produce the vibrato effect. Even after a few weeks I’m seeing progress, and my finger is getting used to the pressure.
Using the wrist as leverage and not the finger itself seems to be key. You finger isn’t strong enough on its own. You need the combined strength of your whole hand to maintain the pressure needed on the string. I also find that letting my left thumb curl over the fretboard helps with the overall process.
And also I’m not worried about the speed of the vibrato right now. I know that will come with time if I get the technique correct. If you give this a try yourself I’d also recommend staying fairly high up on the neck in the early stages where the string is easier to move. Otherwise, you’ll just tear up your finger below the 7th fret.