The University of Oklahoma
BYU and Lavell Edward's Cougarswon the WAC every year from '76 to '85. They also won it in '74. As far as I'm aware, that's the longest streak. Alabama comes in next with nine under Bear Bryant
That would be the Southeastern Conference.
If you're asking whether they have ever played in the Big XII conference championship game, the answer is no. But the conference has only existed since 1996.