I've been wondering how to play vibrato on my violin and I've tried. Every time I try, my whole violin shakes and I don't get that crisp bow sound.
Think 'pivot' rather than 'shake'. Sort of. But get a violinist to show you. Words aren't a good way.
That sounds like you got your violin clamped down by whatever you use for vibrato. Now you know how not to do vibrato.
What vibrato doesn't do is slide around on the string. The contact area of the fingertip changes, the finger itself being supple rather than rigid. Vibrato can come from the hand, from the wrist, from the elbow, the whole arm. Speed and depth and effort depend on just what does what and how connected and supple everything is.
So a good first step is trying to feel the vibrations of the violin and string and bow in your arms. The bowing arm, but also the fingering arm.
Once your contact with the string and the violin and your body is supple and every joint able to give, you have a mechanical connection that actually is capable of entertaining a vibrato.
If your violin shakes, that you may have it in a vice-like grip. That's not just bad for attempting vibrato but will also be pretty bad for smooth position changes.
This is an extremely difficult question to answer in writing without being able to see you playing.
Check out Beth Blackerby from Violin Lab on Youtube. She has extremely detailed videos on vibrato with exercises, troubleshooting and instructions on how to practice. Movement of the violin while using vibrato is one of the problems she covers specifically.