Measuring the "frequency stability" of a signal
Your technique might work, provided you window and so on correctly, but it does not really strike me as the right approach for pitch vibrato for a lot of reasons, mostly that the change in pitch is very small compared to other things which could change the "width" of your spectrum (eg, timbre).

I think you'd be better off actually tracking the fundamental pitch in the time domain using a technique like YIN and seeing how much it varies over time.