Bebout entered politics with his successful write-in candidacy to the state house and was a member of the Democratic Party.

He graduated from Shoshoni High School and started attending the United States Air Force Academy after receiving an appointment in 1963.

He was named to the Prep All-American Basketball team after a poll was conducted of coaches and sportswriters in the United States.

[11] Bebout served as president of the Nuclear Power & Energy Company which performed explorations for uranium.

[15] During the 1986 election Bebout was one of four people to file as a write-in candidate for one of five seats in the Wyoming House of Representatives from Fremont County.

[28][29] Bebout filed to run for reelection as a Democrat and won in the general election after defeating Republican nominee Marlene Brodrick, who had been appointed to fill the remainder of Representative Odde's term.

[34] Edward Lee, the chair of the Fremont County Democratic Party stated that resigning would be "the ethical thing for Mr. Bebout to do".

[46][47] In 1992, Bebout was selected to serve as Minority Whip without opposition after representatives Don Sullivan and Bill Vasey, who instead became chair of the Democratic caucus, declined their nominations for the position.

[48][49] After Bebout became a Republican he lost his position on the Appropriations committee to Representative Bill Bensell and his position as Minority Whip to Representative Wayne Morrow, but Speaker Doug Chamberlain, a Republican, appointed Bebout to serve on the Judiciary and Travel, Recreation and Wildlife committees.

[53] Bebout defeated representatives Harry Tipton and Carroll Miller to become Speaker while Rick Tempest was selected to serve as Majority Leader.

[54] During the 1989 United States House of Representatives special election Roger McDaniel and Bebout served as Democratic nominee John Vinich's campaign coordinators.

[72][73] Bebout was selected as one of three finalist candidates by Republican precinct members of the 26th district who would be voted on by the Fremont County Commission.

The four remaining members of the Fremont County Commission voted unanimously on April 3, to select Bebout to fill the vacancy.

[98] In 1988, Bebout was an initial sponsor of legislation in the state house that would take out a loan that would cost $8.5 million a year in order to building a natural gas pipeline from California to Wyoming.

[99] In 1991, the state legislature voted forty-three to twenty-one, with Bebout being the only Democrat voting in favor, to override Governor Mike Sullivan's veto, which was the first successful veto override in Wyoming's history, of legislation giving an extension of tax breaks for wildcat oil wells.

[100] In 2016, the state senate voted twenty to ten, with Bebout against, against accepting the Medicaid expansion from the Affordable Care Act.

[101] Bebout and Ratliff sponsored legislation in 1989, which called for a constitutional amendment to prohibit the implementation of a state income tax without it being voted on through a referendum.

[104] Bebout supported legislation to declare all same-sex marriages, including those conducted outside of the state, void in Wyoming.

[109][110] The state house voted thirty-three to thirty, with Bebout against, against an amendment to increase the amount of districts in the reapportionment plan from sixty to sixty-two.