Bivins began his racing career in 1964, purchasing a used 1955 Ford. He modified the car by adding a roll cage and initially raced using the snow tires that came with the vehicle.
Bivins first entered NASCAR competition in 1975, participating in two races and achieving a ninth-place finish at Michigan International Speedway. In 1976, he competed for the NASCAR Rookie of the Year title, sharing a shop with Richard Childress in Winston-Salem, North Carolina. He participated in 18 of 30 events that season, securing six top-ten finishes and his career-best result of fifth at the 1976 Richmond 400 at Richmond Fairgrounds Raceway.
The 1976 season saw Bivins contend for Rookie of the Year, ultimately finishing in a disputed second place to Skip Manning. According to Bivins, as quoted in the Emporia Gazette in 2007, “I won the Rookie of the Year. We had a celebration dinner. They turned that around and gave it to Skip Manning. (Manning's sponsor) went to NASCAR and told them if his guy didn’t win the sponsor would pull out.” He competed in eight of 30 events during the 1977 season, earning one top-ten finish, before retiring from NASCAR. Bivins cited frustration over losing the Rookie of the Year crown as a key factor in his decision to leave the series.
Following his departure from NASCAR, Bivins participated in a limited schedule for the ASA before pausing his stock car career in 1980. He resumed racing in 2010, competing in a Modified series. Later, Bivins raced at Lakeside Speedway in Kansas City, Kansas, in the Grand National Class, driving car #5 and winning three feature races in 2018. He transitioned to B-Mods in 2019, 2020, and 2021 while living in Lebo, Kansas, before retiring mid-2021 due to health issues.
Bivins was inducted into the Central Auto Racing Boosters Hall of Fame in 2004 as part of the inaugural class, recognizing drivers from the Kansas City area. He accumulated over 400 wins throughout his career, despite a 30-year hiatus from racing between his first and last race, and won track championships on both pavement and dirt at Lakeside and I-70 Speedway.
After retiring from NASCAR, Bivins established his own construction company and later pursued a career as a professional fisherman. He received support throughout his racing career from his stepfather, Jim, and his mother, Kay Jones. He is married to Claudia Bivins, who actively participated on his pit crew, serving as a video recorder and assisting with car repairs. His son, James Bivins, also contributed to his pit crew for several decades.