It is hard to be relaxed so that you can have a nice vibrato! Be patient with yourself, it takes time to familiarize your muscles with the vibrato motion. Think about what is causing your wrist to tense up. Maybe your left thumb is gripping the neck of the violin. This would hinder the free arm motion required for a lovely vibrato. Try playing a series of notes, periodically pulling your thumb away from the violin and then “letting go” of your thumb to let it fall back in a relaxed position.
Really try to do vibrato in slow motion at first, so you have time to tune into your muscles and tell them to relax. You should be able to control things at a slow tempo. As you get more comfortable, gradually increase the speed of motion, but only as much as you can still control a relaxed arm/hand. I like to remind my students that a baby has to learn to crawl before it can walk or run, it is the same with violin technique. Each new thing we introduce, must be nurtured from the slow and careful baby stages. Best wishes as you practice!