Grissom grew up in Alabama, where his father Wayne sponsored short-track drivers and introduced Steve to the motorsport world from an early age. Grissom balanced racing with being captain of his high school football and basketball teams in 1981, a dual commitment that reflected his broader athletic gifts. He turned down a football scholarship at the University of Alabama to pursue a professional racing career. He joined the Winston All-Pro Series and won the championship in 1985, the following year being nominated for Alabama Pro Athlete of the Year.
Grissom made his Busch Series debut in 1986 in the No. 31 Oldsmobile owned by his father Wayne, finishing thirtieth due to engine failure. He ran part-time through 1987–1988 before moving full-time into the series in 1988, posting four top-tens despite limited sponsorship.
In 1990, he broke through with four wins, including two consecutive, and finished third in the standings behind Chuck Bown and Jimmy Hensley. A single Winston Cup start for Dick Moroso at the Atlanta Journal 500 supplemented that year. A lack of funding curtailed his 1991 title chances, but with backing from Channellock and Roddenberry's in 1992 he returned to form with two wins.
Grissom clinched the 1993 Busch Series championship for his family-owned team by 253 points over Ricky Craven, the most significant title of his career. He won the championship operating his own family team — a notable achievement at a time when well-funded factory-backed outfits dominated.
Grissom received a full-time Cup opportunity in 1994 with Diamond Ridge Motorsports, where he competed for Rookie of the Year honors. Despite qualifying difficulties, he recorded three top-ten finishes and was narrowly edged for Rookie of the Year by Jeff Burton. Diamond Ridge also acquired his Busch operation, and Grissom added two Busch Series wins on a part-time schedule in 1995.
His 1995 Cup season produced four top-tens and a 27th-place points finish. In 1996, he opened the year by winning the season's first Busch Series race in the WCW-sponsored car, but was released from Diamond Ridge mid-season after the Miller 400.
For 1997, Grissom joined Larry Hedrick Motorsports. He opened the year by earning the outside pole for the Daytona 500, and went on to record six top-ten finishes and a career-best 21st-place points finish that season. In 1998, he struggled for results and was released, finishing the year in the No. 96 for American Equipment Racing. That season he was involved in a serious crash during the Primestar 500 where his car slid into the inside wall, flipped onto its roof, and the detached fuel tank ignited a fire after contact with Mike Wallace's car. Grissom was released from the infield medical center approximately forty-five minutes later.
Following the closure of American Equipment Racing, Grissom spent 1999 on fill-in duty across multiple teams, including Carroll Racing, Travis Carter Motorsports, and Hensley Motorsports.
In 2000, Grissom was hired by Petty Enterprises to drive their No. 43 Dodge in the Craftsman Truck Series, finishing tenth in points with six top-five results. The season was overshadowed by a fatal accident: Tony Roper turned into Grissom's truck at Texas Motor Speedway, struck the wall head-on, and died from his injuries the following morning. Grissom also made Cup starts that year, filling in for Kyle Petty in the No. 44 Pontiac.
He returned briefly to the Cup Series in 2002 to replace Buckshot Jones in the Petty No. 44, recording one top-ten finish at Richmond International Raceway in ten starts. Subsequent seasons saw him compete in the Nationwide Series for Frank Cicci Racing and then DCT Motorsports and GIC-Mixon Motorsports through 2004.
Grissom joined Jay Robinson Racing for 2005 in the Busch Series, achieving a 23rd-place points finish while drawing criticism for repeatedly using the Past Champion's Provisional starting spot — a provision reserved for former series champions who fail to qualify on time. He ran selected races for Robinson in 2006 and made brief returns in 2007 through 2009 with various small operations.
Grissom's back-to-back Busch Series championship campaign in 1993 — achieved by a family-owned team operating without large corporate backing — remains a notable example of independent success in NASCAR's developmental series. His son Kyle Grissom later raced in the ARCA Racing Series. Grissom's father Wayne, who had supported his son's career since its earliest days, died on August 21, 2021, at the age of 86.