Goddard's racing career began in the United Kingdom in the early 1980s. He competed in Formula Libre events with Eddie Jordan Racing in 1982 and finished third in the 1984 Esso Formula Ford Championship. He progressed to British Formula 3, where he took third in the National Class in 1985 driving a Solar Racing Ralt RT3-Toyota, claiming one win and eleven podiums across the season. He went on to compete in the British Touring Car Championship in 1988 and 1989, winning the 1600cc class at Donington Park in 1989.
Goddard also served as Chief Instructor at the Silverstone Motor Racing School.
A notable administrative milestone came in 1992, when Goddard became the first driver of the modern Formula One Safety Car at the British Grand Prix at Silverstone — a race at which the car was present but not required to enter the track. He returned to Safety Car duties at the 1993 European Grand Prix at Donington Park and again at the 1993 British Grand Prix at Silverstone, where the Safety Car was actually deployed on track. At the time of these appearances he was running as part of the Porsche organisation.
In 1994 Goddard moved to Asia to assist in developing the Formula Asia Championship. He relocated to the Philippines in 1996 to manage the Shell Toyota Alabang teams in Formula Toyota and the Corolla Cup, winning both championships that year. In 1997 he was asked to return to manage and drive for the Total Toyota Alabang Team in Formula Toyota, leading the championship before an injury forced him to miss the final round.
Goddard moved to China in 1999 to run the Ghiasports Racing Team across multiple series including China Formula 2000, finishing second in the AF2000 Championship standings in both 1999 and 2000.
His competitive peak in Asia came in 2002 when he won the Asian Formula Three Championship driving for his own team with Castrol backing — the Castrol Ambi Pur Team Goddard — in a Dallara F398/Tom's Toyota. He took eight wins, twelve podiums, and eight pole positions across the fourteen-race season. He followed that title with third-place finishes in the 2003 and 2004 Asian F3 Championship, with the 2004 season run under the Team Goddard-Minardi banner. He continued in Asian F3 in 2006 and 2007 before stepping back from driving.
Goddard founded Eurasia Motorsport, which ran championship programmes in Formula BMW Asia, Formula V6 Asia, and the Asian Formula Three Championship. The team served as a base for Asian junior single-seater development through the 2000s and beyond.