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I don't think it is to do with finger length, just technique and practice. It is a most unnatural thing.
I have incredibly small hands and they've never been a problem with regards to vibrato. For me, vibrato came naturally, I was never shown it nor taught it, but I'm really grateful that it did come that way as I know it can give all sorts of problems when pupils are almost trying to force vibrato into their playing.
If you're finding it tough going, it really is about practicing frequently, but don't let the idea of vibrato become so obsessive that it starts to cramp up your playing by creating tension in the shoulder and arm. With vibrato, finger independence is important - vibrato while keeping all the fingers down is near impossible. Relax, let the entire left arm hang from the fingerborad of the violin by using the arm weight to press the finger onto the string. Ensure that your left elbow is situated so that it is underneath the violin - don't let it lift out to the left side, otherwise you will be effectively shortening your fingers and reducing the stretch of the hand. Keep the wrist away from the neck of the instrument.
Now, without the bow, just try the vibrato movement on the string. Imagine you are massaging just a tiny little spot to and fro - not side to side. Don't let the wrist tense up, the arm weight alone should be enough to keep enough finger pressure on the string. Try this with each finger, and on each string. You will probably find your weakest finger for vibrato is the little 4th finger. You will need to strengthen this finger a lot, especially if you tend to be a violin student who avoids using the fourth finger unless absolutely necessary. (There are plenty that do!!)
Once you feel more comfortable with the vibrato action, pick up the bow and play some long notes, adding a little vibrato.
Vibrato comes in many forms; fast and wide, slow and wide, fast and narrow, slow and narrow, plus many combinations of the above depending on what expression you are putting into the music. But you need not worry about that at the moment, it will come in time and with practice.
One final word of advice - for once you've got the hang of it - vibrato of the same type on every single note you play can sound as boring as using no vibrato at all.