Check the throttle cables on the handle to make sure they are plugged in. Be sure the wires are connected from the throttle, auger and starter handles to the motor. If any of them are unplugged, the snowblower will not work.
Remove the oil filter from the side of the motor with an oil filter wrench. It may have leaks that are preventing the engine from getting oil. The filter may be cracked. Check for oil leaking around the filter.
Locate the fuel line, a large black hose running from the gas tank to the motor, on the side of the motor. It may have a leak. Run your hand along the hose, feeling for any leak.
Connect a tire pressure gauge to the tires. Most tires need air pressure between 30 and 35 lbs. Check for any leaks in the tire as well.