Why would the brake pedal shake?

Answer 1

Sounds like brake pulsation. Probably from the front brake rotors. Don't cut them, chuck them and get new ones, replace the beat up pads at the same time and adjust the rears-if drum brakes in back.

Brake pads/rotors are usually to blame for vibration during heavey or normal braking

I agree,new pads, rotors, all the way around will help. But if they are bad, you might want to have the rubber lines checked for cracks to ensure future safety.

A pulsating brake pedal which is known in the automotive world as spongy brakes is caused by worn or warped rotors/drums and/or shoes/pads. To correct this problem visually inspect the parts.If the rotors are not perfectly smooth then replace them.