Basically, what if either the Alaska natives or the Inuit managed to domesticate the caribou? Caribou are the same species as reindeer after all, and two reindeer subspecies were domesticated independently, first by Siberian natives, later by the Sami.
I think the Alaska Inuit are probably the most likely to domesticate the reindeer, since their relatives, the Siberian Yupik, had come from Alaska to begin with. Perhaps contact with the Chukchi and other reindeer-herding groups can spur some sort of innovation that can reach Alaska and thus create this revolution in culture.
The local native groups of Alaska (mostly Athabaskan-speaking groups) are probably less likely to develop it except as a transfer from the Inuit. This is logical, since American Indians readily adapted innovations from neighbouring groups (native or European) in due time. They will steal/poach caribou from the Inuit, and eventually form their own herds.
The muskox comes into play with the influence of the caribou on neighbouring cultures. Since a caribou-herding culture would have a noticeable advantage over neighbours, it would either displace them or the neighbouring cultures would adapt it. So in time, the caribou-herding culture will spread to the lands where the muskox lives. From what I've read, I couldn't imagine the muskox not also being domesticated by the caribou-herders. It could be like the Navajo and sheep. This will also spread the range of the muskox throughout the Arctic.
So what are the final effects of this? How much larger can the population of the Arctic be with an evolution to a herding culture? Since this culture, if it arises early enough, will readily spread through the Arctic, the pre-contact population of the Arctic will be far larger thanks to the extra source of food as well as transport (caribou/muskox pulling sleds). And European influence will be fairly rare as OTL, except perhaps to trade muskox pelts/qiviut wool. In the end, Europeans will conquer them (unless Asians do it first, lol), but their culture will be much more nuanced, and the northern natives will be much bigger.
I suppose also the effects on the American Indians south of the Arctic might be noticable. Caribou historically lived in the Rockies and northernmost New England, but expansion of certain deer species and deer parasites helped transmit a deadly disease to caribou that reduced the southern expansion of their range. While extensive deer hunting and cultural knowledge that deer must be destroyed to protect the caribou herd might help make caribou range further south than OTL, it can't possibly be enough (compare the tsetse fly in Africa). However, certain groups would be able to successfully adapt to this and prosper more than OTL, and no doubt the Rockies could have a very resiliant group of natives. So I suppose this would benefit Canadian natives more than US natives.
That's my initial analysis of this idea I had, and I can see there's lots of potential and ways to go with it.