There are some groups in the Plains who were sedentary (or more appropriately, semi-sedentary) agriculturalists. In particular, the Mandan, Hidatsa, and Arikara all farmed along the upper reaches of the Missouri river and tributaries.
That said, even for these groups, bison hunting was still a tremendously important activity, providing meat, hides, and fat. While the introduction of horses significantly improved the ability of some of the other non-agricultural Plains groups to hunt bison they were only just incorporating the new animals into their existing nomadic, bison-hunting lifestyle.
As for dwelling, I'm not aware of the history of tipi's and other mobile shelter in the region, but the Mandan and other agriculturalists on the Missouri river I mentioned previously would construct earthen lodges that were semi-subterranean pit structures with large beam roofs supporting sod exteriors. There are some earlier pit structures, but it seems like this style of household became widely used beginning in the 15th century and then spread from the Missouri river to other parts of the Plains (particularly Nebraska where the Pawnee made use of them). That said, this style of construction was mostly limited to the eastern margin of the Plains and the further west you go the less you see this style of architecture.
Edit: Because of the mobility of bison herds, these earth lodges were usually occupied only during the farming season and then mobile structures like tipis would be lived in the rest of the year that was dedicated to hunting. The majority of the year would however be in these lodges.
Bison hunting on the Great Plains is truly ancient, dating at least to the early Holocene, ca. 10,000 BP. There have been a couple of bison jump sites from this period that involve extinct species of bison (B. antiquus and B. occidentalis). Check out the Olsen Chubbuck site and the Jones Miller site.