Veach began racing at age twelve after being discovered by Dave Fisher, father of former IndyCar Series team owner Sarah Fisher. Within nineteen months he had progressed from go-karts to the cockpit of a Formula BMW machine. Near the end of 2009, while testing a Formula BMW, Atlantic Championship team owner Eric Jensen signed Veach to his team for the 2010 Atlantic Championship season. In early March 2010 the series halted operations.
In his first year in the U.S. F2000 National Championship with Andretti Autosport, Veach recorded ten top-five finishes and four podium results. He entered the series after the season-opener at St. Pete, missing two rounds of race points, yet finished fifth in the driver championship standings. His performances contributed to Andretti Autosport securing the team championship for the 2010 season.
In January 2011, Veach won the 2011 U.S. F2000 Winterfest — a winter race series held in Florida — winning two of the five races and claiming the title by six points over Andretti teammate Spencer Pigot. He also won the 2011 U.S. F2000 National Championship season opener at Sebring International Raceway, ultimately finishing fourth in series points.
Veach made his Star Mazda Championship debut with Andretti Autosport at Infineon Raceway in August 2011, then competed in the series finale at Mazda Raceway Laguna Seca, finishing third. In 2012 he ran the full season in the No. 77 car, finishing tenth in points with a best result of third at Lucas Oil Raceway at Indianapolis.
Veach moved to Indy Lights with Andretti Autosport in 2013, leading the most laps and finishing third at the Milwaukee Mile and winning pole at Auto Club Speedway on his way to seventh in points. In 2014 he collected three wins and nine podiums in fourteen races, ending third in the standings behind Gabby Chaves and Jack Harvey.
After missing the 2015 season due to an injury requiring hand surgery, Veach returned to Indy Lights in 2016 with Belardi Auto Racing. He suffered a mechanical failure while leading the opening race in St. Petersburg, then went on to collect three wins and several further podiums across the eighteen-race season. During his Indy Lights career, Veach finished in the top ten in 43 of the 44 races he entered — a rate of 97.73%.
Veach made his IndyCar debut at the 2017 Honda Indy Grand Prix of Alabama as a last-minute replacement for the injured J. R. Hildebrand. He closed the gap to teammate Spencer Pigot in each session: from 1.5 seconds in first practice to 1.1, then 0.8, then 0.5 in qualifying. Starting nineteenth, he ran as high as fourth before finishing nineteenth — the last car on the lead lap.
Veach competed at the 2017 Indianapolis 500 for A. J. Foyt Enterprises, placing 26th after retiring on lap 155 with mechanical issues.
On September 11, 2017, Andretti Autosport announced a three-year contract for Veach to drive the team's fourth IndyCar full-time for the 2018 season. On September 23, 2020, the team announced Veach would leave ahead of the Harvest GP.
For the 2021 season, Veach was signed by Vasser Sullivan Racing to drive a GTD-class Lexus RC in the IMSA SportsCar Championship, partnering with Frankie Montecalvo.
Veach was named to CNN's list of "Intriguing People" in May 2010 and became the national spokesperson for FocusDriven. He attended the taping of The Oprah Winfrey Show's No Phone Zone program on April 30, 2010, where he met Jennifer Smith, founder and president of FocusDriven. In May 2010 he campaigned among professional drivers for the No Phone Zone, and on June 4, 2010, announced he had added 33 Indianapolis 500 drivers to the initiative, including Danica Patrick and Hélio Castroneves. In July 2010 he released an anti-texting app called urTXT, which automatically responds to received text messages. On September 21, 2010, United States Secretary of Transportation Ray LaHood honored him at the 2010 National Distracted Driving Summit. Veach also released the book 99 Things Teens Wish They Knew Before Turning 16, presented on NBC's The Today Show on March 2, 2011.
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