You didn’t provide power vent water heater’s brand and model but the principles of operation are pretty much the same for all of them.
When the WH’s thermostat detects water temperature dropping below its setting/ calling for heat, a valve is open allowing gas to flow into a solenoid valve. Pressure building up inside the solenoid valve closes the gas pressure switch, activates the relay and the power vent motor.
Once the motor is activated it builds up the air pressure inside the air pressure switch closing the switch, which activates the solenoid valve, opens the gas flow into the burner and ignites it.
The drawing below shows that basic sequence.
So, there might be a few reasons for the water heater’s power vent to act the way it does:
Malfunctioning gas pressure switch
Thermostat out of calibration
Malfunctioning / defective air pressure switch
Defective solenoid valve
Defective blower relay
One of the first three is probably most probable cause but without testing those parts it’s difficult to be 100% sure. There is also a possibility of a clogged / obstructed water heater vent pipe which results in inaccurate pressure sensing by the air pressure sensor.
Except for the PVC vent pipe
which you might be able to check for obstructions (depending on its configuration / accessibility), the rest of the items will most likely require a plumber.