Can one of the capacitor plates vibrate under the action of a sound wave as a sensitive element of a microphone?

Yes, you can. This is how a condenser microphone works. A necessary part of such a microphone is a resistor connected in series with a capacitor. A change in the capacitance of a capacitor under the action of a sound wave causes a change in its charge, and an alternating voltage arises across the resistor, which is supplied to the amplifier.
In modern microphones, the capacitor plates are made of a new material that can store a charge for a long time.

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