Alright I'm back after being too busy in previous rounds to really submit, lol. I think this question has 4 gimmes: Beartic, Cryogonal, Vanilluxe, and Sawsbuck-Winter. What ties all 4 of these mons together in a theme that isn't "Ice-Type / cold / wintry"? Nothing!
So uhhh, default "I grew up at the top of an icy mountain and this is the team I ended up with" gaming it is.

Our Ice-Type Pokémon for the event. Not much to say here, I considered excluding Vanilluxe because I associate the family with being urban (due to Cold Storage), but that's actually the only urban environment they're encounterable in the rest are rural.

Just in case it still needs a justification, cold Sawsbuck goes brrrr (because it's cold, geddit). Also, this is kinda the only one that Sawsbuck's seasonal gimmick properly works for. You could sort of argue Fairy-Type for spring literally just for being pink, Fire-Type for summer (but any of the Sawsbuck are a great fit for Fire-Type in Unova IMO, arguably spring is even better for Fire-Type), and like Ghost-Type for autumn? Maybe? But with the Ice-Type association it's much clearer than those, because A) Ice-Type has the monopoly on being cold, and B) winter is p much the most important part of the seasonal cycle, Sawsbuck needs the winter phase in order to survive as a Grass-Type otherwise its leaves would keep growing to the point it can't stand up.

This very good borker rescues people stranded in blizzards, and has such a fuzzy wuzzy warm coat that any human sleeping next to it is warm and cozy even in the coldest winter. Ultra Moon notes that " A long time ago in cold regions, every household kept a Stoutland. " Well consider me a trainer from a cold part of the region, and I guess I have a Stoutland (let's just say Ultra Moon happens long enough after BW do to call it "a long time ago".

OK yeah this one seems strange, but my gut felt like it fit and it actually really does. As early as B2W2, Volcarona's dex entry reads " Thought to be an embodiment of the sun, it appeared during a bitterly cold winter and saved Pokémon from freezing. ", and by Scarlet it even says " Its burning body causes it to be unpopular in hot parts of the world, but in cold ones, Volcarona is revered as an embodiment of the sun. ". So Volcarona is more popular in colder territories, making them more livable for humans, and its mythology therefore largely relates to its actions in snowy and wintry places. Considering we're not making a villainous team leader whose MO is to freeze the entire region to death *cough* Ghetsis.2.0. *cough* Volcarona is actually a wonderful fit for "grew up in a freezing cold climate, here's the team I got".