Lots of things.

I presume the engine starts and runs. I also presume the Blower is turning and mechanically functioning. If so, then the engine is basically not taking a "load (i.e. the snow)." I'm also going to presume engine compression is OK.

Most probable cause is a fuel/carb related issue. Check:

-for old, contaminated gas and replace. This is very common.

-fuel line internal wall for deterioration. Small bits of rubber get into the carb inlet, needle & seat, and screw up the float operation, or get deeper into carb and block small orifices. Also check fuel tank for sediment & stuff. This is why fuel filters are always a good idea.

-the spark plug. If it's dark the engine is running rich, possibly because float is "hung" or set too high. A rich running engine will bog down/stop under load. (low engine compression will also do this). Good idea to simply replace the spark plug too, especially if you can't remember when it was last changed.

-If it has an air filter, remove it and see if the engine runs better.

If the above basics don't fix it, you probably have a carb rebuild in your future. These types of carbs are good to learn on, but if any of the above suggestions sound "foreign" to you, take the unit into a shop.