Not a full answer to your question since I can't really speak to clothing and shelter, but one method for storing food for the winter would be the creation of pemmican in large quantities. Pemmican was a mixture of animal fat, meat, and wild berries and nuts that could be stored as bricks or bars wrapped in animal hides for a long time without spoiling. High energy, this was a common food across the plains (and Great Basin) for any time of shortage, not just winter. Hunting is also still possible during the winter, with the aid of snowshoes which were developed independently in North America from those of the Old World.

Keep in mind also that the ancestors of the various groups living in the Plains (and all Native Americas, for that matter) crossed into the Americas from Siberia during the end of the Pleistocene in an ice age and had to have already developed successful strategies for dealing with cold winters and food scarcity. Keep in mind then that they did have more than ten thousand years to adapt to the environment and develop successful survival strategies for surviving cold weather situations.

Many of the Northern Great Plains groups used earth lodges. According to this site,the earth lodges were the primary winter residence of the Arikara, Mandan and Hidatsa, and many other northern Plains groups. Earthen lodges were constructed by excavating a shallow pit then constructing a wooden framework that would support a combination of sod, earth and grass. The sod, earth and grass provided an excellent insulation against intemperate weather. Semi-subterranean houses were used throughout the west in area where winter weather was severe. They were very effective. I discuss the use of plateau pithouses,that employ many of the same principles of construction and insulation here. However, Great Plains earthen lodges were enormous compared to Plateau pithouses.