Born in Conowango Valley, New York, he made his name driving stock cars in the region. When NASCAR introduced its "strictly stock" series, Rexford appeared in three races that were held in the Northeast, with a best finish of 3rd.
He became the first of two Northern drivers to regularly drive in the series in 1950, when it became known as the Grand National division, the other driver being his teammate, Lloyd Moore. That year, he won his first career NASCAR race at Canfield, Ohio. By the end of the year, Rexford managed to involve himself in the race from the championship. He would end up benefitting from the misfortunes three other contenders. Two contenders, Fireball Roberts and Curtis Turner, went through major late season slumps at the same time that Rexford was at his best. A third contender, Lee Petty, had more controversial problems, as he was stripped of over 800 points (at that time, the equivalent of winning 4 races or 5 third place finishes) by NASCAR for racing in non-NASCAR sanctioned races. At the final race of the year in Hillsboro, Rexford had a slight lead and battled with Roberts for the championship. Rexford nearly cost himself the title, when he had an early engine failure. Roberts was able to win a championship with a top 5, but he drove aggressively and tried to win. His own engine gave out with less than 50 laps to go, giving Rexford a dramatic, and controversial championship.
Following his championship, Bill returned to racing in the Northeast in 1951, making a handful of appearances at some of NASCAR's bigger races in the southeast and a few northeastern Grand National events. His win at Canfield was his only career victory.
Over 50 years later, Bill is still the youngest driver to win a championship in what has become the Nextel Cup series. He remains the only driver from the Northeast to win a championship, and was the only Northerner to win a title in the series until 1992, when Wisconsin-born Alan Kulwicki won. Rexford was the only series champion that was not included among NASCAR's 50 Greatest Drivers in 1998.