I didn't find any tutorial on how to do a vibrato with an acoustic piano. So: How to achieve a vibrato with a piano ?
You can do a sideways strong "vibrato" movement (like on a cello, just sideways) on a pressed key. It will depend on the construction of the piano whether any of that tension ends up in the frame where it can in theory affect resonance. The effect will not be a "vibrato" as such but rather a very subtle change in, well, something in the tonal quality. Whether any of that actually makes it to the listener other than via the performer's impression is anybody's bet.
For electronic keyboards or digital pianos, it would be easy to declare that no effect can come from there, but indeed more expensive keyboards actually can record sustained key pressure as "aftertouch" and make it affect the sound, though any such change would hardly be anywhere like the very subtle (if at all) effect on an acoustic piano.
Any such effect is subtle and not dependable enough that I'd be surprised if any serious piano school would bother mentioning it. On the other hand, I'd be surprised if there wasn't a deluge of partly unconvincing clips about it on the video platform of your choice.