Veen was considered one of the top prospects for the 2020 Major League Baseball draft.[5][6][7] He was selected ninth overall by the Colorado Rockies.[8] He signed with the Rockies on June 24 for a $5 million bonus.[9] Veen did not appear for the Rockies organization in 2020 due to the cancellation of the minor league season because of the COVID-19 pandemic.[10]
Veen made his professional debut in 2021 with the Fresno Grizzlies of the Low-A West.[11] Over 106 games, he slashed .301/.399/.501 with 15 home runs, 75 RBI, 27 doubles and 36 stolen bases.[12] He was assigned to the Spokane Indians of the High-ANorthwest League to begin the 2022 season.[13] He was promoted to the Hartford Yard Goats of the Double-AEastern League in early August. Over 126 games between the two teams, he compiled a slash line of .245/.340/.384 with 12 home runs, 67 RBI, and 55 stolen bases while being caught nine times.[14]
He played in the 2022 Arizona Fall League, where he batted .356/.467/.466, and led the league in stolen bases (16; while being caught twice).[15]
Veen spent the 2023 season with the Double–A Hartford Yard Goats, playing in 46 games and hitting .209/.304/.308 with 2 home runs, 24 RBI, and 22 stolen bases. On June 21, 2023, it was announced that Veen would undergo season–ending surgery to repair tearing and fraying to a ligament in his left hand.[16]
Veen split the 2024 campaign between the rookie–level Arizona Complex League Rockies, Spokane, Hartford, and the Triple–A Albuquerque Isotopes. In 65 appearances for the four affiliates, he slashed .258/.346/.459 with 11 home runs, 35 RBI, and 21 stolen bases. Following the season, the Rockies added Veen to their 40-man roster to protect him from the Rule 5 draft.[17]