I received a question recently from one of my Complete Vibrato Mastery students asking whether one of type of vibrato (wrist, arm, or in modern times, the lesser used "finger" vibrato) is the best, and also if it would be ok to not focus on any of these types, but rather, just try to make the sound and see how that goes.
First off, the best vibrato is the one that works best for the music and the individual playing. Some violinists or violists gravitate towards a wrist vibrato, while others find an arm vibrato the most reliable. In the Complete Vibrato Mastery course I mention how many artists use a hybrid vibrato, and also how many use different types of vibrato for different musical situations. I feel that as long as the player is getting the vibrato sound he or she is after (record yourself frequently for objective feedback!), it is consistent (pitch, cycle rate, etc.), has stamina, and is flexible (can be modified to be faster, slower, wider, and more narrow) then yes, it’s important to do what feels most natural and appropriate. While there is a general vibrato sound considered acceptable (consistent both in quality and finger to finger, not too fast or too slow, too wide or too narrow, and definitely not one which makes a player sound out of tune!), to some degree, vibrato is one way in which an artist asserts his or her own personality and style. With that comes a bit of wiggle room, pun intended!
However, it is not uncommon for players to fall into traps when just doing what they feel like doing. What we know about violin playing and pedagogy really hasn’t changed much since the 1700s, when violinist Francesco Geminiani wrote his book The Art of Playing the Violin, which is still just as relevant a read today as it was well over two hundred and fifty years ago! I feel it is incredibly important to develop a technique which feels natural to the individual player, but just as someone can get used to a slouched posture and feel like it's impossible or feels strange to stand or sit with a healthy posture, I think that sometimes we can mistake what we think is natural for motions and actions which are far from it. Especially with such a long and developed tradition of violin and viola teaching and playing, and copious resources and methods for achieving both a natural and highly functional vibrato, there is really no need to struggle alone.
Examples of unhealthy attempts at vibrato might be tensing the muscles and creating a spasm or a "shivering" to get the vibrato motion. Vibrato should never feel tense and its motion should be fairly effortless beyond an initial impulse to get the rocking going. Another pitfall might be collapsing the wrist and rocking with the palm against the neck. This can create a vibrato sound, but intonation and vibrato consistency will almost inevitably suffer. Other common issues are excessive and inconsistent wiggling from various parts of the left arm and hand without a steady pulse, vibrating in ways which detract from the ability to play in tune or to play quickly, and rocking side to side across the strings rather than up and down in the string direction (this later issue is quite terrible for the wrist and creates an inconsistent vibrato sound at best). Finally, some students just want the vibrato to come so badly that they force it to happen without developing and learning to control the motion and end up with something that is usually either too fast, too tiring to ever apply to music in any usable way, or makes them sound out of tune.
If a student is patient and follows the basics of a vibrato method with a teacher close at hand to help circumnavigate bad habits, experimentation is great!