The “foot on the gas” aspect makes one think of CV joints, but symptoms leading to CV joint failures generally involve noise more than the car vibrating.
My vote is the the ball joint at the base of the strut, with a few other suspension components that have some wear on them, allowing a resonant frequency vibration to get set up. This is unusual for a Toyota with only 125k miles, but I don’t know how many pot holes you have hit.
Whatever it is, if it is loose enough to cause the whole car to vibrate, you definitely want to get it resolved before something comes completely apart at highway speed. Click and Clack cannot afford to loose any listeners.