Danner was considered a top prospect at both pitcher and catcher heading into the 2017 Major League Baseball draft. He was selected as a catcher in the second round, 61st overall, by the Toronto Blue Jays.[6] On June 25, Danner signed for a $1.5 million bonus, and was assigned to the Gulf Coast League Blue Jays on June 28.[7][8] He appeared in 34 games, and hit .160 with two home runs and 20 RBI.[9]
Danner did not play in a game in 2020 due to the cancellation of the minor league season because of the COVID-19 pandemic.[11] In 2021, Danner moved to the mound, appearing in 25 games for the Vancouver Canadians of the Northwest League. In 352⁄3 innings, he posted a 2–1 win–loss record, 2.02 ERA, and 42 strikeouts.[9] On November 19, 2021, Toronto added Danner to the 40-man roster to protect him from the Rule 5 draft.[12]
Danner began the 2022 season with the Double-ANew Hampshire Fisher Cats. Danner was optioned to the Triple-A Buffalo Bisons to begin the 2023 season.[13] In 23 games for Buffalo, he registered a 3.81 ERA with 35 strikeouts and 1 save in 28+1⁄3 innings pitched. On August 11, 2023, Danner was promoted to the major leagues for the first time.[14] He made his MLB debut that day against the Chicago Cubs, but retired only one batter before departing the game with an apparent injury.[15] Danner retired Seiya Suzuki on a flyout and reached a full count on the next batter, Yan Gomes, before departing with an injury. The subsequent pitcher Jay Jackson gave up a double on the next pitch. The double was charged to Jackson meaning that Danner is credited with only facing one batter. The next day, he was diagnosed with a left oblique strain and placed on the injured list.[16] Danner was transferred to the 60–day injured list on September 1, ending his season.[17]
Danner was optioned to Triple–A Buffalo to begin the 2024 season.[18] He did not appear for Toronto in 2024, making 33 appearances out of the bullpen for Buffalo, in which he compiled a 3–0 record and 3.15 ERA with 34 strikeouts and 7 saves across 34+1⁄3 innings pitched. Danner was designated for assignment by the Blue Jays on December 20, 2024.[19]