He did not sign with the Yankees and enrolled at the University of Nebraska–Lincoln to play college baseball for the Nebraska Cornhuskers.

[2][4] After signing with the White Sox, Bummer made his professional debut that same year with the Great Falls Voyagers where he compiled a 2.45 ERA in 22 innings pitched.

In 2016, he pitched for the Arizona League White Sox, Great Falls, and Winston-Salem Dash where he was a combined 1–2 with a 4.86 ERA in 15 relief appearances between the two teams.

Bummer spent the remainder of 2017 with the White Sox, compiling a 1–3 record and 4.50 ERA in 30 relief appearances.

[8] In 2021, Bummer appeared in 62 games with an ERA of 3.51 in 56.1 innings while striking out 75 batters as the White Sox made the postseason.

On November 16, 2023, the White Sox traded Bummer to the Atlanta Braves in exchange for Michael Soroka, Jared Shuster, Nicky Lopez, Braden Shewmake and Riley Gowens.