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?