Does A.C. make the material through which it is flowing vibrate?

Oscillating does not always mean vibrating. Oscillation simply means some measurable value is cycling back and forth. This could be a vibration, which would be a measurable change in position, back and forth (like a grandfather clock or your phone's vibrator), but oscillation is a more general concept. For example, in linguistics, we talk of oscillation in linguistic patterns which seem to go back and forth, back and forth. There's no physical movement in that case, but there's still an oscillation.

AC does not automatically mean vibrating. It's merely the fact that the measurable current is oscillating between forwards and backwards. Technically, electrons are moving, but you'll find its a very tiny effect.

However, you will find things operating on AC do in fact vibrate. This is due to second order effects. The changing current can cause changing magnetic fields, which can draw components together and push them apart in tune with the AC oscillations. This is the source of the humming you will hear from many devices.

Some components such as resistors, most (but not all) capacitors, and semiconductors aren't very prone to vibration. Other components like transformers are and have to be constructed to prevent audible vibrations. Back when CRTs were very common, it was not unusual for the coil(s) in their flyback transformer to loosen over time and cause a high-pitched noise. It could be downright annoying for those who could hear it.