The New York Yankees selected Webb in the tenth round, with the 314th overall selection, of the 2013 Major League Baseball draft. The Yankees assigned him to the Staten Island Yankees of the Class A-Short SeasonNew York–Penn League after he signed, but promoted him to the Charleston RiverDogs of the Class ASouth Atlantic League two weeks later. Over relief innings, he went 3-1 with a 3.31 ERA. Webb began the 2014 season with the Tampa Yankees of the Class A-AdvancedFlorida State League, and was promoted to the Trenton Thunder of the Class AAEastern League in May, with whom he was named an Eastern League All-Star. Webb was promoted to the Scranton/Wilkes-Barre RailRiders of the Class AAAInternational League following the All-Star break. For the season, Webb went 3–6 with a 3.80 ERA and 12 saves and 94 strikeouts in 48 games and innings pitched. Webb received a non-roster invitation to spring training with the Yankees in 2015, and opened the season with Scranton/Wilkes-Barre. Due to a tendon injury, he only pitched 38 innings during the season, and the Yankees assigned Webb to the Arizona Fall League after the regular season. In 2016, he pitched for Scranton/Wilkes-Barre, going 4-3 with a 3.59 ERA over innings pitched out of the bullpen. On December 8, 2016, the Pittsburgh Pirates selected Webb from the Yankees in the Rule 5 draft. He competed for a role with the Pirates during spring training, but did not make the 25-man roster and was returned to the Yankees before the regular season. Webb began the 2017 season with Scranton/Wilkes-Barre. The Yankees promoted Webb to the major leagues on June 22, 2017, and he made his MLB debut on June 24. He made seven relief appearances for the Yankees, pitching to a 4.50 ERA while striking out five and walking four in six innings.
Milwaukee Brewers and San Diego Padres
On July 13, 2017, the Yankees traded Webb to the Milwaukee Brewers for Garrett Cooper. Following the trade, Webb pitched for Milwaukee and the Colorado Springs Sky Sox of the Class AAA Pacific Coast League. Over two innings with Milwaukee, he gave up two runs, and over 17 games with Colorado, he went 1-2 with a 6.48 ERA. Webb was designated for assignment on April 7, 2018. On April 14, 2018, Webb was claimed off waivers by the San Diego Padres. He made one appearance for the El Paso Chihuahuas of the PCL before the Padres promoted him to the major leagues.
St. Louis Cardinals
On June 29, 2018, Webb was claimed off waivers by the St. Louis Cardinals and assigned to the Class AAA Memphis Redbirds of the PCL. He was recalled by St. Louis on July 27. In innings pitched for St. Louis, he compiled a 1.76 ERA. Webb began 2019 with Memphis. He was recalled by the Cardinals for the first time in 2019 on April 7, and optioned back to Memphis on July 11. He was recalled on July 21, finishing the season with St. Louis Over 65 relief appearances during the regular season, he went 2-1 with a 3.76 ERA, striking out 48 over 55 innings.
Personal life
Webb's father, Kirk, pitched in college baseball for Old Dominion University. Prior to Webb's first promotion to the Yankees, he had never been to a major league game.