Sounds like it could be a problem with the condenser. This is usually part of the ignition points assembly. If it is new enough to have electronic ignition, the ignition module could be failing.

Either way, you should be able to still get parts for it.

The ignition system of a small engine is usually located under the flywheel. You may need a puller to remove the flywheel. If it has electronic ignition, you will need to replace the ignition module. If it has points, replace the points and condenser. These ignition parts can usually be purchased as a tune-up kit.

You should be able to get the parts from a small engine repair shop. Make sure you have the make, model, and serial number of both the snow blower and the engine.

If you are not very mechanically inclined, you may want to spend the extra money and have the repair shop install these parts for you.

If it has a kill switch, make sure that it's not vibrating to the off position.

You could have a carburetor problem, a sticking float will cause it to use up the fuel in the float bowl and then die until more flows in. If your engine has a fuel pump, it could be weak. Either way make sure the pickup screen in the tank is clean, a slow trickle of fuel will fill the float bowl, allow you to start and run for a minute, but can' t keep up with the demands of full throttle.

You may even be getting some icing in the carburetor.