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.
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.