Refer to figure.
The figure represents the pitot-static system on a very simple aeroplane with only the basic instruments. This is typical of what you will fly when obtaining your PPL. The vertical speed indicator is designed to show your rate of climb/descent. It does this by measuring how quickly the static pressure is changing - this acts as a reliable representation of how quickly the altitude is changing.

The VSI works similarly to an altimeter, except it has a small intentional "leak" between the capsule and housing. If the static pressure remains the same, the pressure equalises between the capsule and the housing and the needle reads zero. However, as the external pressure changes, the pressure in the capsule changes before the pressure inside the housing - and the needle will indicate a climb or descent.

So, the VSI only requires static pressure, and only uses the static port. This is the same for the altimeter. It does not require any electricity from the electrical system, or vacuum pressure.

The ASI (Airspeed indicator) requires both static and total pressure, to calculate the dynamic pressure. So the ASI uses the static port and pitot tube.