I overfilled engine oil in my Briggs Straton engine on my snow thower ... now can't get it started .... can oil foul the spark plug ?

Hi and welcome to FixYa, I am Kelly. Depending on just how much you over-filled it yes it will fowl the spark plug. Most of the time though a small engine failing to start is either fuel or ignition related. Remove the spark plug and check for ign. spark and OIL in the cylinder by pulling the recoil through. Reinstall the spark plug (Just a note about an oil fouled spark plug.... if you clean it with fuel you must make sure that it is dry or it will not work. Spark plugs are cheap I recommend you buy a new one.
Try removing the air filter and adding just a small splash of gas into the carb. throat. then see if it tries to start. If it does but does not run you have a fuel supply problem. This is caused by debris in the carb bowl and fuel orifice jet. You will need to clean out the bottom of the carb and the orifice. Possibly there is debris in the fuel tank outlet, the fuel filter or fuel bowl below the fuel selector depending on your engine model. .I can help you with carb / fuel system cleaning procedures but would like a "complete" ENGINE model - type - and serial number so I can pass you the procedures applicable to YOUR engine.

Thanks for choosing FixYa,
Kelly