I was learning vibrato on the violin, and came across this video. In this video, it tells you that vibrato is just a fast slur. Using that method, how would you play vibrato on an open string?
There are two main things you can do.
First, you can finger the pitch an octave higher on the next string and do vibrato on that. The note you're fingering will vibrate sympathetically from the overtones, so this will actually have an effect.
Second, you can do the motion of vibrato with your left hand, but with your finger somewhere that's not doing anything. The slight shaking of the instrument will produce a variation in intensity. It's not quite the same as true vibrato, but it's something.
Really though, you would try to avoid using an open string for a sustained note with vibrato.
I'm not a violinist, but I think you simply would not do this.
If you want to apply vibrato to an open string, you grab that note on the next lower string and put vibrato on that note.
There are some guitarists who press the string before the nut or slightly bend the neck to apply pitch changes to an open string, but I have never seen a violinist do that.
Contrary to what the previous answers said, it is actually possible to play vibrato on open strings: just set your finger right on the nut† and vibrate with pretty much the normal technique, so that the upper half of the swing puts some slight pressure on the string.
The resulting vibrato isn't optimal intonation-wise because the deviation is only upwards, whereas classical schooling recommeds that vibrato actually swings more downwards. But whether this is a problem depends on the concrete musical piece. For what it's worth, the standard vibrato technique on steelstring guitar also only changes pitch upwards.
†Not behind the nut, but actually on the nut. Behind the nut is again a technique for steelstring guitar, that relies on increasing the tension of the string while keeping the length constant. The technique I'm describing keeps the tension constant but varies the effective length, like a normal proper fretless-string-instrument vibrato. This wouldn't work on guitar because the vibrating down from the nut strongly damps the string in addition to raising the pitch, which would basically squelch the note. But on violin, the bow sustains the sound so this is not a problem.
Dredging up ancient memories of my violin and viola-playing days. If you are playing an open G,D or A string then the trick is to hold down the note one octave higher on the next string (3rd position) and use vibrato on that note. You don't sound this note with the bow, and you don't achieve as vivid effect as that of a bowed, fingered note, but the effect is still audible.
In general I was encouraged to never play an open string unless it was absolutely necessary.