I would love to hear some string teachers address their sequences for vibrato. My own (still evolving) is...

1) Polishing the string
2) Polishing the string while the thumb stays in place
3) Slowly applying pressure until the left hand finger is rolling instead of sliding
4) Practicing at different speeds with a metronome.

Step 2 is a big one for students to make and takes a lot of individual instruction to keep the wrist from flexing. Is there an extra step that would make my students more successful?

Vibrato can be hard. Check out this book, Playing The String Game by Phyllis Young. She has some great suggestions on fun games to play to get your students on the right track.

I tell all my students that they need a few prerequisites regarding how they hold their instrument before vibrato will work at all.

With vibrato, there are 3 points of contact from the instrument to the body.

1. Thumb to neck of instrument (thumb MUST be loose and not "grasping" the instrument)
2. The finger being vibrated (this finger must be tall, and only the tip of finger on the instrument)
3. The chin/shoulder on the instrument.

Different kinds of vibrato work for different players (and teachers): finger, wrist, or arm depending on what joint is moving. Of course cello and bass vibrato is a different motion than violin and viola vibrato.

As Kate said, there are prerequisites to vibrato including good posture and generally playing in tune using altered fingerings.

You may want to try this as beginning motion. Play a pitch in tune and then bend it back a quarter step flat. After this motion gets easier then play a pitch, bend it a quarter step flat, and then return it to pitch.

There is debate as to where the pitch actually resides during vibrato, but most as I do, hear the correct pitch at the top of the oscillation, not the middle or bottom.

Rotate this exercise through all fingers on all strings. Then begin to add rhythms and play scales with bending the pitch back and returning to pitch with gradually faster rhythms, and as you said, metronome markings.