The matter of the coexistence of horses and humans in North America is interesting, although it branches away from the typical, written history you see on much of this subreddit. Most of what we understand is just inferences based on what was going on ~10,000 years ago when they went extinct. I am primarily a geologist, although I was trained as a historian in college, so understand my perspective from this light.
The number that is often given, around ~10,000 years ago, for the extinction/extirpation of horses from the continent is well-respected and largely based on the fossil record (Guthrie 2003, Haile et al. 2009, Waters et al. 2014). Horse fossil dwindl and then disappear around that time, while others persist. This is generally how we know if a species has become extinction. Animals, especially large ones occupying a large range covering multiple geologic processes (difference stages of deposition or erosion), will leave behind fossils. Some studies have used environmental dna that is stuck in the permafrost to identify what animals were living in an area at a certain time (this is outside of my specialty) and those have also seemed to confirm the extinction of North American horses around this age (Haile et al. 2009).
Geologists, biologists, and anthropologists seem to have migrated to the idea that much of the Pleistocene megafauna (Horses, huge Bison, Mammoths) may have gone extinct from climate change, human predation, or a combination of the two (MacFadden 2005). Climate change in the later quarternary in North America has been linked to a large vegetation shift, which has been supported by Paleoclimate studies and pollen records (Anderson 1988). This idea of climate changing leading to their extinction is somewhat supported by the body size decline shown in the horse fossils preceding their extinction (Guthrie 2003).
However, there seems to be a growing wave of literature which is linking the megafauna extinctions to humans. The global extinction history of the megafauna was compared to climactic change and human paleobiogeography (Sandom et al. 2014). This study strongly linked the severity of megafauna extinction with arrival of humans around the same time. This can be even further supported in our North American story at Wally's Beach in Canada, where there is a site where horses and camels were being butchered around 13,300 years before present (Waters et. al 2014).
While I have not really answered your question related to an oral history of horses in modern native communities, I hope I have helped to provide some information on the interaction between early North Americans and horses. It seems like their relationship was one of predator-prey, and the North American horse may have been hunted to extinction (or the hunting of other species changed the environment/vegetation regime to cause extinction).
Sources
Anderson, P.M., 1988. Late Quaternary pollen records from the Kobuk and Noatak River drainages, northwestern Alaska. Quaternary Research, 29(3), pp.263-276.
Dale Guthrie, R. Rapid body size decline in Alaskan Pleistocene horses before extinction. Nature 426, 169–171.
MacFadden, B.J., 2005. Fossil horses--evidence for evolution. Science, 307(5716), 1728-1730.
Haile, J., Froese, D.G., MacPhee, R.D., Roberts, R.G., Arnold, L.J., Reyes, A.V., Rasmussen, M., Nielsen, R., Brook, B.W., Robinson, S. and Demuro, M., 2009. Ancient DNA reveals late survival of mammoth and horse in interior Alaska. Proceedings of the National Academy of Sciences, 106(52), 22352-22357.
Sandom, C., Faurby, S., Sandel, B. and Svenning, J.C., 2014. Global late Quaternary megafauna extinctions linked to humans, not climate change. Proceedings of the Royal Society B: Biological Sciences, 281(1787), 20133254.
Waters, M.R., Stafford, T.W., Kooyman, B. and Hills, L.V., 2015. Late Pleistocene horse and camel hunting at the southern margin of the ice-free corridor: Reassessing the age of Wally’s Beach, Canada. Proceedings of the National Academy of Sciences, 112(14), 4263-4267.