will it cause vibration? yes,if its a loose point between the tire and steering wheel. that will always cause a vibration.

making the brakes pull to the side? not as likely, unless it is really worn, pulling if you eliminate all possibilities in the braking system can be control arm bushings, ball joints, strut mounts, bearing or tie rods. basically something is allowing the spindle to shift while braking.

But most likely it can be traced to something in the braking system.