I would start by removing the plate and inspecting the wiring. Start by turning off the breaker that controls the light. It is possible that some of the wires are touching or perhaps there is a lose wire on one leg of the switch. Normally, this type of situation would trip the breaker.Have you noticed any flickering of the light itself? You might consider replacing the bulb to see if that has anything to do with the vibration.

If this is a single pole switch both legs on the switch should be black plus a green ground. All other wires in the box should be wirenutted off.

Thank you - I will add it to my list for the electrician.

Is there a refrigerator or other appliance on the other side of the wall? Do you feel the vibration only on the plate, or on the wall also? Is the plate metal or plastic? Is it a regular switch or a dimmer?

I suspect your face plate is magnetic (made of iron based material). Check it with a magnet. If it is magnetic, that may be your problem. A current carrying wire laying close to the plate will induce an alternating field in the plate and hence vibrate.

The other possibility is a dimmer control mounted to the face plate.

Most hum sources in household lighting circuits come from magnetic components such as chokes, relays, and transformers. A broken shading coil on a relay magnet will can turn it into a buzzer. In fact, I had washing machine with a buzzer that it was nothing more than a standard relay without its shading coil. Terminals and contacts were ommitted also.

There is no appliance on the other side of the wall.

The switchplate is a heavy brass Baldwin plate.

Magnets stick to it but metal paperclips do not.

There is a bank of 6 more switches above it with a similar brass metal switchplate but it is not Baldwin. There is no vibration on the 6 switch plate.

There is no dimmer.

The vibration occurs only when the switches are on.

It is a double switch and occurs when one or both are on.