I just heard that Spanish explorer Juan de OƱate describes sled teams of shaggy dogs on the plains. This was obviously before the proliferation of horses. What do we know about the use of dogs? Did this practice continue with horses or disappear? Did any of the woodland cultures use sled dogs?
If you'll allow me, I can go just north of the Rio Grande here.
The Karankawa, a nation of five or so related tribes along the Texas gulf coast between Houston-Galveston and Corpus Christi and centered on Matagorda, were known for their affinity towards dogs. In fact, it is supposed that the name "Karankawa" translates as "People who love dogs" in one of the seven neighboring Coahuiltec languages. This was, at least, the explanation offered a few centuries ago - they called themselves "Keles" which was simultaneously the verb for "to wrestle", and the noun "wrestler" in both singular and plural form. So, while they might not have considered themselves especially into dogs, the people around them most certainly made that association. Perhaps with good reason.
The Karankawa were nomads. The Gulf Coast, like the Great Plains, featured a wide and open prairie that would later see use in ranching. Now, especially along rivers and bayous, this could also condense into thick, almost jungly swamps - many places on the Gulf Coast easily meeting the requirements to be classified as a rainforest. As well as this, they traversed the calm coastal waters to reach barrier islands seasonally. To traverse these sorts of habitats, the Karankawa used a mixture of two methods. The first was their use of long dugout canoes for riverbound and coastal travel, while using sort of crude sleds while on land. Like many other nomadic nations, these were more often than not pulled by dogs.
The dogs of the Karankawa were quite peculiar. According to European observers, they highly resembles coyotes. They were called "silent dogs", because they did not often bark or howl. The Karankawa used them not just for pulling sleds, but also as hunting dogs, and ALSO as simple pets like any modern doggo might be.
Some sources suggest that these actually were coyotes, taken as puppies and given all the proper taming and socialization that they needed to be effective work animals and pets. While this is a strange story, it's also not entirely out of the question. Coyotes have, in some special cases, shown an ability to be social and friendly in a very domesticatable fashion. Quite a few have interacted with humans in relatively positive or passive ways, some even have gone begging to people - something that even wolves raised in captivity show reluctance to.
So, what are they? Well, they're pretty much coyotes. These aren't really the only sled dogs of North America, not by a long shot. They are, however, a peculiar situation for the Karankawa, one of the nations that had been nomadic since before European contact and had remained nomadic long after. Although it was said they made excellent horsemen when horses were introduced, they never adopted them for the use of pulling sleds. So, the practice of dog sleds pulled by sled dogs continued regardless of European interference and the injection of a new animal to the Americas.
I can't say exactly why it never changed. One might suppose there was some special feature of a dog sled that a horse cart simply could not keep up with. They were relatively lightweight, ideal for a nomadic lifestyle, and were fully capable of hauling the limited cargo a band, clan, or tribe might carry. In more northerly regions, a dog might be simply more adapted to the cold than a horse might be. The sleds also function better in wet, marshy coastland and rainforest, and perhaps snow as well, since they float and glide and lack any wheels to get stuck in mud, rocks, ice, or frost. Above all, they were already a thing, so it would've been an active effort to establish horse carts in place of what had sufficed for so long. This much is all my speculation, though.
If you wish to see a sled in action, that picture was drawn around the upper Missouri River in the 1830s. It shows that, in many ways, dog sledding has not changed much at all these past few hundred years.
Sources:
Cannibal Coast by Edward Kilman, 1959
Karankawa Indians: The Coast People of Texas by Albert Samuel Gatschet, 1891