A standard pressure operated VSI admits static pressure into a belows, with a capilliary tube "leaking" that pressure into an airtight instrument case.
If the static vent were partially blocked, then the instrument would underread, if the orifice on the capilliary tube were partially blocked, then the instrument would overread.
The Flight Data Recorde on the 737 at least is powered from the No. 1 Transfer Bus and the Battery Bus. On the ground it starts to record when either engine oil pressure switch exceeds 35 psi - in the air it records regardless. I would gess that the DFDR would start recording on engine start.