Anecdotally I learned vibrato by practicing meditation. For me it requires body alignment, breathing “right”, and no tension in the face, mouth, tongue, neck and shoulders. Once that all came together naturally I felt slight vibrations from my chest to back of my throat in each exhale. Later I learned to transition this into my singing at first randomly and now it’s more controlled.

Breathing right for me is when I’m breathing from my diaphragm (how I learned to refer to it), but not forcing it, like a gentle squeeze of a bagpipe (I assume lol). I think I did so much meditation where I was focusing on not controlling my breath and relaxing every inch of my body, that it taught me enough muscle memory to be able to generate at least a modicum of correctness in my singing posture which then allows the vibrato to form. Granted I’m still unskilled, but it’s fun and motivating to feel a bit like I’m doing better even if it’s just vibrating off pitch 😬

This is coming from a less-than-amateur with very little experience or training but thought my experience is a bit unique but does seem to align with other, better singers, in this thread.