Try anchoring with the web of your hand between your thumb and pointer finger, maybe. The “blues” hand position with your thumb over/around the neck definitely lends itself to a wider “shake the string around” sort of vibrato than the “classical” thumb-behind-neck position.
Actually, scratch that, I just grabbed my strat to see what I actually do when giving a note some vibrato, and it depends a bit on which finger is engaged (mostly in terms of an inxreased hand angle) but it seems the two contact points are my thumb, around the last joint, on the top/low E side of the neck, and then the “ball” where my pointer finger joins my hand on the bottom/high E side of the neck, and that the vibrato is driven mostly by wrist rotation.
Yngwie’s vibrato isn’t JUST gear, but certainly his preference for light strings tuned down a half step, and a neck that allows you to get a good “grip” on the string rather than on the fretboard below it (scalloped, jumbo frets, whatever), certainly helps… Some of it is definitely technique and practice too, being ale to grip and move the string with little resistance helps get you a wider vibrato, but having the control to do it at a rate that feels musically appropriate, rather than at hummingbird/bumblebee tempos, is also pretty critically important, too.