Veijer began riding minibikes from a young age, claiming the Dutch Minibike Championship when he was just eight years old. His progression through junior categories was methodical: he entered the Red Bull MotoGP Rookies Cup in 2020 and finished tenth in his debut season, while also competing in the European Talent Cup that year. A second Rookies Cup campaign in 2021 saw him finish twelfth, hampered by missing two rounds at the Red Bull Ring.
His final two junior seasons proved decisive. In 2022 he finished as runner-up in the Red Bull MotoGP Rookies Cup with three victories and six podiums, while simultaneously competing in the FIM JuniorGP World Championship, where he took two wins and three total podiums. These results established him as one of the most promising Dutch prospects in two-wheel motorsport.
Veijer joined the Moto3 World Championship in 2023 with Liqui Moly Husqvarna Intact GP, partnering experienced campaigner Ayumu Sasaki. His debut season showed measured growth: in only his tenth race, at the Austrian Grand Prix, he started from pole position for the first time and ultimately finished fourth.
Consistency across the second half of the year brought tangible rewards. Veijer earned his maiden podium in Thailand, then secured his first World Championship victory in Malaysia, crossing the line ahead of teammate Sasaki and title rival Jaume Masià. He closed the season seventh in the standings with 149 points. Although he narrowly missed the Rookie of the Year award — which went to David Alonso — his late-season surge underscored his pace in a competitive class.
Retained by Intact GP for 2024, now alongside Tatsuki Suzuki, Veijer took a significant step forward. He secured his second career Moto3 victory at Jerez, then backed it with a third place at Le Mans. At his home race in Assen he crossed the line in second after being overtaken at the final corner by Iván Ortolá, a result that resonated strongly with Dutch fans.
Veijer qualified on pole for the German Grand Prix — his second career pole — but crashed while leading early in the race, remounting to finish eighteenth. Despite that setback he gathered multiple additional podiums through the remainder of the year and ended the season third in the championship standings, behind David Alonso and Daniel Holgado. The result confirmed him as a front-runner in the class.
On 13 August 2024, Red Bull KTM Ajo announced that Veijer would graduate to Moto2 for 2025, partnering Deniz Öncü. The step up presented the expected learning curve. His early-season results were modest but pointed in the right direction: a tenth-place finish at COTA in round three was followed by points-scoring outings at Qatar and Jerez.
A training accident injured his right arm and required surgery, forcing him to miss the British Grand Prix and the following round. After returning to action, Veijer progressively rebuilt his form. He scored multiple top-ten finishes in the second half of the season and capped his rookie Moto2 campaign with a maiden podium at Portimão, finishing second by less than a tenth of a second behind Diogo Moreira. He ended 2025 fifteenth in the Moto2 standings.
In September 2025, Red Bull KTM Ajo confirmed Veijer's renewal for the 2026 season, with José Antonio Rueda named as his new teammate.
Veijer represents the continued strength of Dutch motorcycle racing talent, a lineage that has seen the Netherlands produce multiple World Championship contenders. His rapid ascent through the junior ladder and twin Moto3 victories before the age of twenty placed him among the most credentialed young riders in the intermediate class. His ability to recover from injury and still claim a podium in his Moto2 rookie year marked him as a genuine future contender for the intermediate-class championship.