I am from Sootopolis, Hoenn and I recently took a vacation to Alola where I caught an alolan vulpix. Would she do alright living in Hoenn?

Yes, of course. Alola and Hoenn have very similar climates, and although Alolan Vulpix are specifically native to the colder, more mountainous parts of Alola, they fare just as well in the sunnier, more humid coastal areas. Moreover, as Sootopolis benefits from the shadows of the caldera it’s located in, your home may even be cooler than the Alolan coastlines many tamed Alolan vulpix grow up around. Just be sure she’s always hydrated, and brush her daily to keep her fur coat thinned and free from any heat-trapping debris. Otherwise, she should be fine.