After coming up through the junior ranks of Grindavík, Jón Axel played his first senior match with the club in 2011. He won the Icelandic championship with the club in 2012 and 2013 and the Icelandic Basketball Cup in 2014.
In 12 regular season games with Grindavík during the 2014–2015 season, Jón Axel averaged 15.6 points and 6.4 assists per game. He had a rough first round series against eventual champions KR, going scoreless in game two and averaging only a 6.7 points in the three game sweep. During the 2015-2016 season, Jón Axel continued his improvement, averaging a career high 16.8 points and 8.0 rebounds, along with 5.2 assists per game. He scored a career high 35 points against Keflavík on 8 February 2016. In the playoffs, Grindavík faced KR again in the first round. Jón Axel posted a triple double in the first game, with 10 points, 12 rebounds and 10 assist but despite his performance, KR won 85-67. KR blew out Grindavík in the next two games and completed another three game sweep for the second year in a row. For the series, Jón Axel averaged 8.3 points, 8.3 rebounds and 5.7 assists.
College
Jón Axel joined Davidson College in 2016. On December 26, 2016, he was named the Atlantic 10 Conference Rookie of the Week. On November 13, 2017, he was named the Atlantic 10 Conference Player of the Week. On March 11, 2018, Jón Axel won the Atlantic 10 Conference championship with Davidson and helped the team advance to the 2018 NCAA Division I Men's Basketball Tournament. He averaged 13.2 points, 6.0 rebounds, and 5.1 assists per game as a sophomore. Coming into his junior season. Jón Axel was named to the Preseason Third Team All-Atlantic 10. On 17 November, 2018, Jón Axel scored 33 points in Davidson's 57-53 victory over Wichita State in the opening round of the Charleston Classic. On 26 January 2019, Jón Axel scored 27 points, including two free throws with 5 seconds left, in Davidson's 54-53 victory against Saint Louis. On 22 February 2019, he posted a triple-double, 20 points, 10 rebounds and 10 assists, in a 75-66 victory over Rhode Island. It was the first triple-double for a Davidson player since John Falconi had 17 points, 12 rebounds and 11 assists in 1973. On 25 February he was named the Atlantic 10 Player of the Week after averaging 23.0 points, 9.0 rebounds and 7.5 assists during the week of 18-24 February. On 12 March 2019, Jón Axel was named the Atlantic 10 Player of the Year as well as being selected to the First-team All-Atlantic 10 after averaging 17.2 points, 7.3 rebounds and 4.7 assists per game for the season. In April 2019, Jón Axel declared for the 2019 NBA draft. In May 2019, he had workouts with the Sacramento Kings and the Utah Jazz. On 27 May, he announced that he would withdraw from the draft and return to Davidson for the 2019–20 season. In December 2019, Jón Axel was named to the U.S. Basketball Writers Association watch list for the Oscar Robertson Trophy. In 2020 he became the first Division I men's player since 1993 to hit career marks of 1,500 points, 700 rebounds, 500 assists, 200 made 3-pointers and 150 steals. At the close of the regular season, he was named to the Third Team All-Atlantic 10. With the 2019–20 season having been his final season of college eligibility, Jón Axel was automatically eligible for the 2020 NBA draft.
Skyliners Frankfurt
On 15 July 2020, he signed with the Fraport Skyliners of the Basketball Bundesliga.
