Dry swirling snow or a blowing snow can be hard. If you have a prevalent direction, closing vents on the upwind side can really help but that may not eliminate the problem, especially with a dry swirling snow. Shutters or something like that could help.
A roof vent may be your best solution. Maybe you can find something on Craigslist or maybe you have one of those recycling places for building materials. If you have much snow, I would not recommend a ridge vent. It could get blocked. Or maybe you could build some type of cupola.
Ammonia is also a risk from the poop. Ammonia is lighter than air so you need something over their heads when they are on the roosts so it can get out. Warm air holds more moisture than cooler air and warm air rises. Another reason to have something open over their heads.
How important is ventilation? That depends. Your risk with the cold is frostbite, not them freezing to death unless they are weak to start with or maybe you have really bitter cold, way below zero Fahrenheit. If you are where it is cold enough so the poop stays frozen, not much moisture or ammonia is going to come out of their poop until it thaws. A reduction in ventilation is not nearly as big a problem until the poop is thawed.
Another thing that can reduce the risk is to keep the poop cleaned out of the coop. If it’s not in there it can’t give off moisture or ammonia.
Many people have posted on here that they have solved frostbite problems by providing more ventilation. It can be really important.
I sometimes get snow or rain inside my coop. Not often because my opening where this comes in is on the non-prevalent wind side. I have a walk-in on the ground and use wood shavings. They get wet but pretty quickly dry out since I have a lot of ventilation. Just stir the shavings up or throw some corn in there so the chickens will stir it for you. How well this works for you will depend in your climate and how wet it gets.
Your best solution may be a roof vent or cupola so you always have some ventilation, along with closeable vents you can open up in better weather, especially hot weather. Heat is usually more of a threat than cold.