Can bad motor mount cause car to vibrate at idle and cold start?

I have a 96 corolla and when I start up my car in the morning or when its been sitting for a few hours, the car is loud and the right front bumper, side marker light and turn signal light vibrates/makes noise because the bumper, side marker light, and turn signal light is touching each other or things next to it, and thus causing the loud vibration noise. When I hold them with my hand, it goes away. Can the right front/top motor mount be broken to cause this vibration noise?

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msrdude
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