I started to learn vibrato in September. I'm not sure which type it was; it involved pulling my fingers back straight, and then bending them on the tip, bringing the hand more into the violin.
When I do it, my violin tends to move/shake. I don't know if it's the vibrato i'm doing or if i'm not supporting the violin properly? I can hold my violin up comfortably and drop my arm by my side- which I tried when i found this problem. Does anyone have any advice? Thankyou!

Perhaps you are shaking your hand a little too vigourously

You must be grabbing too firmly at the point where your (left) index finger contacts the neck of the violin. Pull your thumb a little under the neck so that there is a 1mm gap and not more. This means you have to support the instrument a bit more with your head and a bit less with your left hand. A shoulder pad is useful here. This will free the left hand to move. The knuckles give and take loosely.

For practice purposes, rest the scroll of your instrument against a wall. This will remove any fear you might have of dropping the instrument.

Good Luck. Let me know if this helps.