I cannot say what the Dakota specifically did, but I can tell you something specific about horses. In his book, the Horse, the Wheel, and Language, which is in large part about how proto-Indo-Europeans domesticated horses and spread out from the steppe of Russia and Ukraine all the way to Spain on the one hand and to Western China on the other, David Anthony talks about how most domesticated animals would starve in a field covered in snow and ice. Horses, however, have some ability not only to recognize that there is grass under the snow but also to use their hard hooves to break through ice.

Horses are easier to feed through the winter than cattle or sheep, as cattle and sheep push snow aside with their noses and horses use their hard hooves. Sheep can graze on winter grass through soft snow, but if the snow becomes crusted with ice than their noses will get raw and bloody, and they will stand and starve in a field where there is ample winter forage just beneath their feet. Cattle do not forage through even soft snow if they cannot see the grass, so a snow deep enough to hide the winter grass will kill range cattle if they are not given fodder. Neither cattle nor sheep will break the ice on frozen water to drink. Horses have the instinct to break through ice and crusted snow with their hooves, not their noses, even in deep snows where the grass cannot be seen. They paw frozen snow away and feed themselves and so do not need water or fodder. In 1245 the Franciscan John of Plano Carpini journeyed to Mongolia to meet Güyük Khan (the successor to Genghis) and observed the steppe horses of the Tartars, as he called them, digging for grass from under the snow, “since the Tartars have neither straw nor hay nor fodder.” During the historic blizzard of 1886 in the North American Plains hundreds of thousands of cattle were lost on the open range. Those that survived followed herds of mustangs and grazed in the areas they opened up.^7 Horses are supremely well adapted to the cold grasslands where they evolved. People who lived in cold grasslands with domesticated cattle and sheep would soon have seen the advantage in keeping horses for meat, just because the horses did not need fodder or water. A shift to colder climatic conditions or even a particularly cold series of winters could have made cattle herders think seriously about domesticating horses. Just such a shift to colder winters occurred between about 4200 and 3800 BCE (see chapter 11).

Therefore, there wouldn’t be a need for the Dakota (or any other nomadic horse people, from the proto-Indo-Europeans to the Mongols) to specifically store hay for winter. It allows nomadic horse pasturage in areas where cattle-herding and shepherding simply isn’t possible. Horses, even before man came into the picture, even before domestication, evolved on the Eurasian Steppe to eat grass even when it was covered in snow and ice in harsh Russian and Mongolian winters. The open prairies of North America apparently proved just as suitable.