For a few years, I have experimented with various strategies and techniques to introduce and develop students' vibrato. On the whole, I feel vibrato is taught too late in string sequencing. The one element that most interferes with students' success is a sub-par shoulder rest. When the shoulder rest is not adequate, the left hand/arm becomes a kickstand more than a functional mechanism for musicality. My students with foam shoulder rests (for example) are least effective in both learning and executing a quality vibrato. Consequently, the mobility necessary for vibrato is locked. Walking around one day helping a student who said, "I just cannot get this!" I slightly elevated their scroll so it was disengaged from the left hand. Suddenly this student was miraculously able to do a fantastic vibrato motion. Illustrating for students the importance of a quality shoulder rest is vital!
We teach students block hand shape to develop muscle memory and fingerboard geography; however, this develops two habits that can be difficult to reverse. The first is the adhesion of the index finger to the neck of the instrument (particularly for violin and viola) that makes vibrato next to impossible. The second is the difficulty associated with keeping multiple fingers down during a vibrato-ed pitch. Most frequently, the index finger is disengaged from the neck and elevated like one is pointing toward the sky. It would seem most effective to codify the mental map of the fingerboard (through the Ionian pentacle on each string) during the first year and teach chromatic alterations and the use of single-fingers during the second year. Disengage the index finger!
Variable speed within the vibrato motion is important, to meet the demands of the music. The standard practice using a metronome to assist in "metering" the motion and increasing speed is a great tool; however, it feels sooooooooooo dull. Throw some Richard Simmons on YouTube for a more entertaining tempo reference - the kids love it. True Story: I have had an entire class of well-respecting students clear the stands and chairs to join in the motions for at least 5 minutes. It was a proud moment in my teaching career.
Another idea is to increase vibrato speed through a crescendo. This trains the hand (and brain) that long notes are not "dying" notes but "thriving" notes. Playing long tones (whole notes) at 60BPM provides a chance to develop "vibrato gears" to shift between. From this you can develop a common language for students to assign vibrato intensity to (1) each piece, (2) sections throughout the piece, and (3) individual notes.
