A state cannot make a law in violation of the Federal Constitution. This means that a state cannot establish a religion.
However, there is nothing in the Federal Constitution that says a state cannot mandate owning a gun, or owning anything else. A state would be perfectly legitimate to pass a law which required its residents to own a gun, insurance, a dildo, steel balls for their truck, a blow up doll, or anything else.
A problem would arise, however, if the SD law says a person must own a gun made by a specific manufacturer, for instance, "everyone must own a Colt .45". That would likely be deemed unconstitutional.