How does an engine driven vacuum pump used in gyroscopic instrument work?
It uses a system of vanes to draw air through the inlet port and expel it through the outlet port. The engine's lubricating system provides oil for lubrication as well as sealing the internals. The inlet is connected to the instrument and this provides the vacuum the instrument needs. To answer your last question, if the instruments are connected in series and then to the vacuum pump, and one of the instruments obstructs the air flow in the system, you may loose vacuum and instruments can become unreliable or stop working.