The fuel pump starts when ignition is in position 2, as the whole system needs a cerain pressure. It will switch off again when the pressure is reached after abt. 5 seconds and when engine is not started.
There is no vacuum sucking the fuel when engine is running, it is the pressure from the fuel pump and that is more than the car needs. Therefore there is a or more (depending on engine type) fuel pressure regulating valve. And there is a return line for the excess fuel to the tank. And there is a check valve inside the tank that the fuel does not leak back into the tank when engine is not running or car is parked.
For the price of the pump you can check online at for example www.autohausaz.com