You will notice a little device on the bottom of the carburetor with wires going to it. That device is the fuel shutoff solenoid. Lock down the brake and then put your hand on that device then turn the key on to see if you can hear/feel it click. If you do, then we know the fuel shutoff solenoid is working. If it is not working you will need to follow the power wires from it all the way back to the ignition switch to see if you can find a break in the wire. If there are no breaks then it's possible the solenoid just burned out (not that common).
If you determine the fuel solenoid is indeed working then I would lean towards there being an issue with the carburetor itself. It's quite possible there is some trash or built up moisture in the carburetor causing the problem. This engine shows to have a carburetor where the bowl is held on by 2 screws and the fuel solenoid. Because of that, I recommend you take the carburetor off to open it up and clean it out. Trying to remove those 2 screws with the carb mounted on the engine is extremely hard to do in most cases as they are in there very tight.
You can pick up a can of spray carb/choke cleaner from any auto parts store to use for cleaning the carburetor out. Make sure to remove the needle and float and spray up into that passage way to clean out any debris that may be present.
If cleaning it doesn't seem to take care of the problem, I advise replacing the carburetor.
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