Bummer participated in the baseball team at Sunrise Mountain High School in Peoria, Arizona.[1] In the 31st round of the 2011 MLB draft, he was selected by the New York Yankees. He enrolled at the University of Nebraska–Lincoln to play collegiate baseball for the Nebraska Cornhuskers instead of signing a contract with the Yankees.[2]

Bummer was selected a league all-star in 2013 while playing collegiate summer baseball with the Cape Cod Baseball League’s Harwich Mariners.[3] As a junior in 2014, he started 15 games for Nebraska and finished with a 7–5 win–loss record and 3.34 earned run average (ERA).

Bummer was chosen by the Chicago White Sox in the 19th round of the 2014 MLB draft, with the 558th overall pick.[2][4] Bummer signed with the White Sox and played for the Great Falls Voyagers in his first season as a professional, pitching 22 innings with a 2.45 ERA. 2015 was a lost year for him owing to injury. He made 15 relief appearances in 2016 while pitching for the Arizona League White Sox, Great Falls, and Winston-Salem Dash, going 1-2 overall with a 4.86 ERA for the two clubs. In 2017, he started the season with Winston-Salem, received promotions to the Birmingham Barons in May and the Charlotte Knights in July.

On July 27, 2017, Bummer’s contract was selected by Chicago. That same evening, he faced the Chicago Cubs in his Major League debut and struck out Anthony Rizzo, the first hitter he faced.[6] Before being called up, he had a 3.31 ERA and a 1-5 record in 49 innings pitched between Winston-Salem, Birmingham, and Charlotte. Bummer played with the White Sox for the balance of the season, recording a 1-3 record with a 4.50 ERA in 30 appearances as a reliever. Bummer made 37 appearances in 2018, logging 31+2⁄3 innings at an ERA of 4.26. He saw a significant improvement the next season, making 58 appearances, tossing 67+1⁄3 innings with an ERA of 2.13, and striking out 60. He inked a five-year, $16 million contract with the White Sox.