Beckmann began karting in 2008 and collected major victories in 2009, 2012, and 2014. In 2015, he graduated to single-seaters with Mücke Motorsport, competing in both ADAC Formula 4 and Italian F4. He ended that debut season fifth in ADAC F4 (taking the rookie championship) and fourth in Italian F4.
Beckmann moved to the FIA Formula 3 European Championship in 2016 with Mücke Motorsport. Forced to miss the first two rounds due to his age, he debuted at Pau and collected two podiums and a fastest lap, finishing fifteenth overall and seventh in the Rookie Championship. In 2017, he switched to Van Amersfoort Racing but struggled for three rounds before making a mid-season move to Motopark, where he scored 45 points and finished sixteenth.
For 2018, Beckmann joined Jenzer Motorsport in the final season of the GP3 Series alongside Juan Manuel Correa and Tatiana Calderón. After a slow start at Jenzer, he switched to Trident mid-season and transformed his results — taking three race wins and an additional podium to finish fifth in the championship.
In 2019, Beckmann competed for ART Grand Prix in the newly formed FIA Formula 3 Championship. In 2020, he returned with Trident, delivering a stronger campaign with two wins and four further podiums to finish sixth in the standings.
Beckmann stepped up to Formula 2 in 2021 with Charouz Racing System. He showed promise early in the season with a podium on debut at Sakhir and a second place in Baku. Financial difficulties forced him to leave Charouz mid-season; he subsequently signed with Campos Racing to replace Matteo Nannini, scoring a best finish of fifth at Monza, but was again replaced before the campaign concluded, ending the year fifteenth in the standings.
In 2022, Beckmann made four appearances across different teams. He drove for Charouz at Imola, then deputised for Van Amersfoort Racing at Silverstone, Le Castellet, and Budapest before being promoted to a permanent seat at VAR for the remainder of the season from Spa-Francorchamps onward, replacing Jake Hughes. He was himself replaced for the Abu Dhabi finale by Juan Manuel Correa.
Beckmann was announced as the test and reserve driver for Avalanche Andretti Formula E during the 2022 Mexico City ePrix weekend. For the 2022–23 season, he continued in that role at Andretti while also joining Porsche as a reserve driver. His Formula E race debut came at the Jakarta ePrix double-header, deputising for André Lotterer as Lotterer prepared for the 24 Hours of Le Mans.
Beckmann joined Cupra Kiro for the 2024–25 season alongside Dan Ticktum in his first full Formula E campaign, but scored only a single point before parting ways with the team at the end of the year. He subsequently became a reserve driver for both Porsche and Cupra Kiro for the following season.