Oscar Jr. studied at the University of New Mexico from 1944 to 1946, where he also played football. He then went to the University of Utah, where he received a bachelor's degree in 1949 in political science and a Juris Doctor in 1952.
From 1955 to 1957, McConkie was a member of the Utah House of Representatives. McConkie was elected president of the Utah State Senate during his first term in office. It was the first time a state senator in his initial term was elected president of the senate since the beginning term of the Utah State Senate. In 1956, he ran against William A. Dawson for Utah's 2nd congressional district.[9] He also served very briefly in 1965 and in 1966 as acting governor[10] when both the governor and secretary of state of Utah were not in the state.
McConkie was president of the Salt Lake City School Board[10] and from 1982 to 1984 he was chair of the Utah State Board of Education. He was chair of the Governor's Committee on Education from 1983 to 1985.
McConkie was a senior law partner representing the LDS Church.[11][12] McConkie argued the case that led to a broadening of the clergy-penitent privilege in Utah to include any information that was given by a parishioner to a cleric with the intent of getting spiritual advice. McConkie has been involved in international law, having drafted laws that were enacted in both Jamaica and Mauritius.
McConkie was a key figure in the process of getting official recognition of the LDS Church in Zaire in 1986.[13]
McConkie was named lawyer of the year by the Utah State Bar in 2007.[14]
McConkie, Oscar W. Jr. (1962). The Kingdom of God: A Study Course for Priests Under 21 and Bearers of the Aaronic Priesthood Over 21 Years Old in The Church of Jesus Christ of Latter-day Saints for 1963. [Prepared under the direction of the Presiding Bishopric]. Salt Lake City, Utah: Church of Jesus Christ of Latter-day Saints.
—— (1963). God and Man: A Study Course for Priests Under 21 and Bearers of the Aaronic Priesthood Over 21 years Old in the Church of Jesus Christ of Latter-day Saints for 1964. Salt Lake City, Utah: Presiding Bishopric of the Church of Jesus Christ of Latter-day Saints.
—— (1964). The Priest in the Aaronic Priesthood: A Study Course for the Priests and Aaronic Priesthood-Adults in the Church of Jesus Christ of Latter-day Saints for 1965. Salt Lake City, Utah: Presiding Bishopric of the Church of Jesus Christ of Latter-day Saints.