The best time to start learning violin vibrato? Day one.
Often vibrato is considered an “advanced” technique, which should be pursued only after the basic foundation of the left hand is securely in place.
But instead, vibrato is the natural result of a healthy approach to the instrument.
With that in mind, I ask my beginning students, young or old, to perform motions that resemble (and in fact are) integral to a relaxed and mature vibrato. These motions emanate from both the wrist, elbow and hand joints, which all play a part in the final result.
Some players may want/need to wait before making a formal study of vibrato, but in fact I’ve seen many very young players in their first year of study pull it off with an impressive amount of finesse and freedom.
For me the bottom line is why wait? Developing and perfecting the vibrato reflex takes months and even years. It is integral to every style of playing, and so it should be addressed not as an “add-on” but from the very beginning as an essential part of playing.