Snowblower engine do not have air filters. Moisture around the hot engine would freeze up on the air filter element, choking off the engine. Also, there is no need for a filter, as you don't have the dust or dirt in the snow.
Lots of engines run fine without a hot box in snow conditions. Lawn tractors with a snowblower mounted on front works fine with a air filter on the engine. The hood is acting as somewhat of a hot box to keep the carb and the linkage warm but snow still gets in there. The engine can take the snow blowing into it on a conventional snowblower, it will miss and slow down at times and need a breath of clean air sometimes but most of the time recovers and keeps going. Snow Kings were famous for connecting rod break even when full of oil because of the hot box working to well at times. The owner in a big hurry shut it down and did not clean it off and let it run for several minutes. When he returned hours later and started it the engine over revved due to linkage frozen in place or butterfly frozen in place and the engine was history. Some guys also used to take the box off because the engine ran smoother and was easier to reach the throttle linkage to pull a string and run the engine faster under heavy loads to blow snow faster and better. The linkage would freeze in place and when the string was released the throttle did not spring back, the engine over revved and once again it was history. The air filter thing not being on a snowblower is a mixed blessing. The engines are still ingesting dirty air as one can see from filthy snow the day after a storm. My dark blue truck will be grayish in only a few days from nasty dirt on it from the air after a snowstorm. But they do just keep on running in the worst of conditions. I have many times had to stop and clean my face of ice but the Snow King just kept on going.