Groff made his professional debut in 1984 competing in the Valvoline/Robert Bosche Formula Super Vee series. He finished third in the series in 1985 and second in 1986. In 1986 he also made his American Racing Series debut (later renamed Indy Lights). Driving for a family team, he captured wins at the Milwaukee Mile and Road America and finished third in points. In 1987, he won at Nazareth Speedway and finished fifth in points. In 1988, he finished second at Nazareth while competing in three Indy Lights races for three different teams. In 1989, he competed full-time for Leading Edge Racing and captured four wins on his way to the championship.
In 1990, Groff attempted to qualify for the Indianapolis 500 for Euromotorsport but failed to qualify. He made his debut three weeks later at the Detroit Grand Prix with the same team and completed the season with them. In twelve starts he had a best finish of seventh at Nazareth and finished seventeenth in points. In 1991, Groff initially returned to Euromotorsport but switched to A. J. Foyt Enterprises for the final five races of the season. His best finish was eighth (three times) and he finished sixteenth in points. He also competed in his first Indianapolis 500. In 1992, Groff qualified a second car for Walker Racing in the 1992 Indianapolis 500 but when regular driver Scott Goodyear failed to qualify, he replaced Groff. He made six starts that season for Euromotorsport (two races), Walker (one race), and Foyt (three races) and finished 23rd in points. In 1993, Groff joined Rahal-Hogan Racing for a limited race schedule. Groff made six starts in 1993 for Rahal-Hogan with a best finish of ninth at Portland International Raceway. He finished 23rd in points. In 1994, Groff returned to Rahal-Hogan full-time. The team struggled with Honda’s new CART engine package throughout the season and had to purchase Ilmor-powered cars from Team Penske to make the 1994 Indianapolis 500. Groff finished twentieth in points.
In 1995, during qualifications for the 1995 Indianapolis 500, Chip Ganassi Racing tabbed Groff to drive the back-up car for Bryan Herta, who had crashed and was not cleared to drive during the weekend. Groff started a qualifying attempt, which was aborted after two laps. Herta was not bumped, and Groff did not drive during Bump Day.
In 1996, Groff joined the new IRL IndyCar Series with Foyt Enterprises. Groff finished sixth in the series' first race and third at the second race at Phoenix International Raceway. He also raced in the CART race at Nazareth with Walker, finishing fourteenth, and qualified eleventh for the 1996 Indy 500 with Walker, but was sidelined by a fire after 122 laps. Groff finished sixth in the 1996 IRL championship. For the 1996-1997 Indy Racing League season, Groff joined Byrd/Cunningham and captured three straight top-five finishes to start the season including a third at the Las Vegas Motor Speedway and a second at Walt Disney World Speedway. He finished twelfth in the 1997 Indianapolis 500 and finished fourteenth in the championship. Groff returned to Byrd-Cunningham Racing in 1998. After finishes of seventh, fifteenth, and fifteenth to start the season, he was replaced at the Texas Motor Speedway race by John Paul Jr. He was out of a ride for the rest of the season. Groff attempted to qualify for the 1999 Indianapolis 500 for Team Xtreme Racing but failed to make the field. It would be his last IndyCar appearance.