As Fixdit said, depending on how old the engine is of course, it should have the solenoid on the bottom of the carby. The solenoid's purpose is to prevent the engine 'running on' after the ignition is turned off.

As Norm also said, the ignition shouldn't be the cause as it uses a magneto.

See the excerpt from the B&S workshop manual below:

Testing Fuel Shut Off Solenoid
The fuel shut off solenoid is controlled by the
equipment ignition switch. With the switch OFF,
the solenoid plunger closes, stopping the fuel
flow at the fixed main jet. With the switch in the
ON and START positions, the solenoid plunger
opens, allowing normal fuel flow. If operating
properly, the solenoid will click when the switch
is turned ON and OFF.

If convenient, can you post the engine model, type, & code? It should be etched onto the top of the rocker cover.

The red arrow in the picture below is pointing to the solenoid.
[Linked Image]