shaking while braking is because the rotor(s) is warped

rotor warping could be caused by many things, such as external force, defective material which deforms over time, rotor uneven wear, or other factors. replacing warped rotors is not uncommon on cars more than 100kmi. Roter self is inexpensive, about $20 apiece for most non-luxury cars. if you can have somebody diy, it should be an easy fix.

in your case, you said only front and rear bumper have minor scratches/dents, i assume the force of impact was small - otherwise the bumper (nowadays almost all jap's cars are hollow plastic - you didn't say what kind of car you are talking about though) which is very unlikely causing rotor warping.

you can safely deposit the check to your bank account and get a fix for the rotor. it's not a big deal. good luck.