I know this is a silly question, but I couldn't help but wonder it. Is there a way to play vibrato on the piano? If not are there pianos out there with this feature?
Vibrato is impossible on the piano. The closest you could come is with the growling effect using the damper pedal, but this can be risky.
Vibrato is a standard technique on the clavichord, where it is called Bebung. This is possible due to the direct mechanical connection to the tangents which strike the strings. This possibility was a casualty of the development of the piano's escapement mechanism.
In the era of extended techniques, nothing is truly impossible anymore. At the expense of a nice sound and precise pitch, you can change the pitch of the note you are playing by pressing a finger or a suitable object on the strings. Obviously, if you wiggle it around you get a sort of vibrato.
There are quite some modern pieces that require altering the pitch of strings like this, however I haven’t heard any that require vibrato.
Touching the strings with your fingers may cause them to rust.