Webb played at the high school level for Prosper High School in his hometown of Prosper, Texas under coach Kent Scott. While at Prosper High School, Webb guided the team to a district championship and a trip to the Texas 4A regional semifinals.
Webb earned several accolades and was named first team All-District as a junior and honorable mention All-State. Webb graduated from Prosper after passing for 2,658 yards, 589 yards rushing, and 36 touchdowns.
Webb was ranked a 4-star prospect coming out of high school by ESPN and signed with the Texas Tech Red Raiders. During his freshman year in 2013, Webb alternated as a backup quarterback behind fellow freshman Baker Mayfield and as a starter, becoming only the second true freshman at the position to ever play for the school. Webb had been afflicted by an undetermined illness, and had lost 40 pounds in several days prior to the start of the season.
Webb made his first game appearance in the contest against TCU, where he threw the game-winning touchdown to wide receiver Bradley Marquez. Following an injury to Mayfield, Webb made his first career start in an appearance against Iowa State, which saw him pass for 415 yards and 3 touchdowns. His performance marked the fourth most yards passing for first time starter in school history.
He would follow up his Iowa State performance with a record-setting performance against West Virginia. Webb passed for 462 yards, 36–50 attempts, and two touchdowns, making Webb the first quarterback in school history to pass for over 400 yards twice in their freshman season. Additionally, the yardage broke the school freshman passing record previously held by Billy Joe Tolliver. The passing yardage also set a Big 12 Conference freshman passing record, beating the mark set by Baylor's Nick Florence in 2009.
The game versus Oklahoma State saw Webb complete 45 passes on 71 attempts for 425 yards. The number of completions and attempts ranked fourth overall, and second overall behind Texas Tech quarterback Graham Harrell.
Webb struggled in an outing against Kansas State, and did not play in the second half. However, his 78 yards thrown pushed him over the previous record for the school single season freshman passing yards previously held by Zebbie Lethridge in 1994. After Webb's poor performance against Kansas State, Baker Mayfield would return to the starting role for the next two games against Baylor and Texas.
Following the conclusion of the regular season and during preparations for the 2013 Holiday Bowl, it was announced that fellow quarterback Baker Mayfield would transfer from the school following a 'miscommunication' and the open competition for the starting position. The departure of Mayfield ensured that the only competition for the starting job for the Holiday Bowl would be Michael Brewer.
After the month of bowl practice, Webb was named the starter for the Holiday Bowl against a top 15 Arizona State team. Webb would provide one of the most impressive performances of the 2013–14 NCAA football bowl season. He finished the game with 403 yards, 28–41 passing, 4 touchdowns, and no interceptions in the 37–23 upset, tying a Holiday Bowl touchdown record and earning MVP honors.
Webb would conclude his freshman season with several Big 12 Conference freshman records: single game total offense, single game offensive plays, most 400 yard passing performances, passing yards in a single game, passing attempts in a single game, passes completed in a single game. His performance also marked school freshman records for completions, attempts, touchdowns, passing yards, and 400 yard games. Webb would later be named honorable mention Freshman All-American by College Football News, Big 12 Broadcasters Conference Freshman of the Year, as well as several weekly honors.
Entering into the 2014 season, Webb was named the starting quarterback following the transfer of fellow quarterback Michael Brewer to Virginia Tech. Both Webb and the team as a whole were named dark horse candidates for the Heisman Trophy and the national championship respectively by ESPN. Webb was also named to the Maxwell Award and Manning Award watchlists for the best quarterback in college football. Webb started the first eight games of the season before suffering a season-ending ankle injury in a game against TCU on October 25, 2014. He was replaced by true freshman Patrick Mahomes. It was later revealed that Webb would undergo surgery on his shoulder, which he originally injured on September 25 in a game against Oklahoma State.
In 2015, Webb was the backup quarterback to Patrick Mahomes, who retained possession of the starting quarterback job. On December 30, 2015, Webb announced his intentions to transfer from Texas Tech.
On January 27, 2016, Webb announced his intention to transfer to the University of Colorado Boulder. At the time of the announcement, Webb was still enrolled at Texas Tech and was nine credit hours short of finishing his degree requirements. After finishing his degree requirements and graduating from Texas Tech, Webb planned to be in Colorado for the first summer session. In late May, Webb revealed that he would be transferring to the University of California, Berkeley, instead of Colorado. On May 25, 2016, California Golden Bears head coach Sonny Dykes announced that Webb had signed a financial aid agreement and is officially enrolled. As a graduate transfer, he was eligible to play immediately.
Webb was drafted by the New York Giants in the third round, 87th overall, in the 2017 NFL Draft.