If your engine only runs for a few seconds and then shuts off, you could have a gummed-up carburetor. The engine probably starts because gas enters the cylinder when you press the primer bulb. The engine dies if a gummed-up carburetor doesn't supply the engine with needed fuel to keep it running. Rebuild or replace the carburetor if the engine dies soon after start-up.
These repairs may help solve your gas snowblower problem
Clean and rebuild the snowblower carburetor
Rebuild kits are available for most carburetors. The kit contains the essential components such as jets, pins, seals and gaskets to overhaul the carburetor. You can sometimes fix a fuel supply problem by disassembling and cleaning a carburetor. Use the rebuild kit to rebuild the carburetor after cleaning it.
Over time, varnish from stale gasoline builds up in the carburetor, decreasing the amount of air/fuel mixture reaching the engine. Replace the carburetor if it's severely clogged with varnish or if the orifices are clogged with debris and can't be cleared using carburetor cleaner and compressed air.