What you're describing sounds like "disk brake chatter". It's a condition caused by warped rotors. Rotors often warp when they're old and/or have been overheated.
Try replacing brake pads.
Your front brake rotors need to be machined or replaced.
The Squeal means its time to replace the Brake pads.
front brake pulsation have front rotors replaced
The Vehicle shakes because of the virbration produced by the engine at the stop sing
my first thought is the brake pads are warn out. pull the front wheels check the pads and rotors. your question was car stoped foot on brake in drive, CAR SHAKES i would say that the car is missfireing, bad plug ? bad wire? bad coil pack? or mabey a BURNT VALVE? Good luck I just had that problem and I replaced the master brake cilinder and it stop shaking. try that. it shakes because of the anti lock brake system.
Defective or misadjusted stop light switch?
Startoff and brake to stop slower
I'd start with the disk brake rotors. If one or both is warped it could cause the problem you describe.
possible warped brake pads or warped rotor Shaking while stopped, can be caused by broken motor mount, or engine miss.
brake relay or sensor
The brake pedal has a hinge or pivot pin, give it a shot of lubricant,should stop squeaking.