Veach began a motor racing career in 2008 in karting. In 2010 he moved to the Formula racing. Originally it was a start was planned in the Atlantic Championship, however, the series before the start of the season driving one. Instead Veach Andretti Autosport rose to the second event of the U.S. F2000 National Championship in the Championship one. With a second place and three podium finishes he finished the season in fifth place in the drivers' standings, while his teammate Sage Karam dominant won the championship. In 2011 he took first for Andretti Autosport in the winter series of the U.S. F2000 National Championship and won it with two wins in five races. He then competed in the regular championship. In this he won the season opener and finished the season on top of the fourth. Internally, he lost his teammate Spencer Pigot, who finished second. In addition, Veach Andretti Autosport in 2011 debuted with in the Star Mazda Series. He took as a guest part in two races and once reached the third place.
Early 2012 initially decided Veach the winter series of the Star Mazda Series for itself. He then went for Andretti Autosport as a regular pilot in this series. With a second place as best result he reached the tenth overall, while his teammate Karam total was third. 2013 Veach received an Indy Lights Cockpit at Andretti Autosport. With a third place he once stood on the podium. While his team-mate Carlos Muñoz was third with four wins, Veach finished the season in seventh place in the driver standings.
- 2008: Karting
- 2010: U.S. F2000 National Championship ( 5th place)
- 2011: U.S. F2000 Winter Series (Master)
- 2011: U.S. F2000 National Championship ( 4th place)
- 2012: Star Mazda Series, Winter Series (Master)
- 2012: Star Mazda Series (No. 10)
- 2013: Indy Lights ( 7th place)