I don't see any obvious reasons why that couldn't work.
Fuel, as noted by some other answers, could be an issue if your people live entirely on the treeless tundra, but if they spend any time at all (perhaps only winters) in or near the taiga, they'll have plenty of wood to burn for fuel. Reindeer (which, even if not domesticated, were traditionally one of the main sources of meat and furs for arctic hunter-gatherers) live in both regions, often migrating between them, so it's perfectly reasonable for your arctic people to follow them.
As for "wasting" heat that could by used for staying warm (pointed out as a potential problem in some other answers), the obvious solution is to build your forge inside the hut or tent or dugout that you're living in. You're going to be heating that space anyway, so you might as well get some useful work out of it. Build your forge with a big pile of rocks to store the heat, fire it up in the evening, work some metal while you're waiting for the rocks to heat up, and finally douse the fire and let the rocks keep the place warm all night while you sleep.
As a bonus, you can also use the stored heat for cooking food, and the same structure can also serve as a sauna for washing up. Minus the forging, that's pretty much how a traditional Finnish savupirtti ("smoke cottage") worked, and I see no reason why something similar couldn't work for your people, too. Of course, you don't have to combine all those functions into one structure, but if heat and fuel are scarce, that's one way to optimize it.
The biggest issue I see with your scenario, however, is economic: if the metal is not produced locally, why would traders bother to haul it all the way up north as raw ingots, when they could probably (especially early on, before your arctic smithing tradition was established) make a much better profit hauling finished goods instead? Traditionally, metal was usually either worked locally close to where it was mined, or it was traded to cities where smiths and other craftsmen would gather. Your proposed system, with scattered smiths in the arctic working on traded metal, kind of runs counter to that pattern.
That said, I don't think that's an insurmountable issue. Certainly many places historically had village smiths, specializing in simple repairs and adaptations of existing tools. It doesn't seem entirely implausible for some basic metalworking skills to become a useful and established trait among your arctic population, especially if trade in metal goods was common but irregular (so that metal tools would be reasonably common, but not always easily replaced if they broke), making the ability to repair them useful.
Getting the necessary skills introduced in the first place could be an issue, but that could plausibly come down to a one-time event, maybe even a single trained smith fleeing to or being captured by the arctic folk. And once established, it's entirely plausible that your arctic folk might develop some unique small-scale metalcrafting tricks or traditions that would justify exporting their produce, at least as specialty items or curiosities. Instead of importing raw metal, they might still (mostly) import simple metal tools and weapons and rework them (for practical reasons, or just to decorate them), but that could still allow the maintenance of a local smithing tradition.