Flockhart started competing in 1951 in a JP Formula 3 car. He purchased the ERA R4D from Raymond Mays and in 1953 had a particularly successful season, outpacing one of the works BRMs at Goodwood and achieving podium finishes at Goodwood, Charterhall, Snetterton, and Crystal Palace, alongside several hill climb victories.
His World Championship Formula One debut came at the 1954 British Grand Prix in Prince Bira's Maserati 250F, a shared drive that was not classified. His best World Championship result was third place at the 1956 Italian Grand Prix in a Connaught Type B, which earned him five championship points. He drove for a variety of teams across the World Championship including Maserati, BRM, Connaught, Cooper, and Lotus, making his final World Championship start at the 1960 United States Grand Prix in a Cooper T51 entered by Cooper Car Company.
In 1959, driving a BRM P25, Flockhart won the Lady Wigram Trophy in New Zealand and also qualified on pole, set fastest lap, and won the non-championship Silver City Trophy.
Flockhart's most celebrated achievements came at Le Mans with the Scottish privateer team Ecurie Ecosse. In 1956 he co-drove an ex-works Jaguar D-type to victory alongside Ninian Sanderson, giving Ecurie Ecosse its first Le Mans win. The following year he returned with the same team and won again, sharing with Ivor Bueb. The 1957 victory set a distance record of 2,732.8 miles (4,398.0 km), a remarkable achievement for a privateer outfit running year-old Jaguar machinery.
Flockhart continued to compete at Le Mans in subsequent years. He returned for Ecurie Ecosse in 1959 alongside John 'Jock' Lawrence in a Tojeiro, but retired with overheating, and again in 1960 with Bruce Halford in a Jaguar D-type, retiring with a crankshaft failure. His final Le Mans appearance came in 1961 for Border Reivers, co-driving an Aston Martin DBR1 with Jim Clark before retiring with clutch failure. His 1955 appearance, sharing a Lotus Mark IX with Colin Chapman, also ended in retirement.
In the early 1960s, the United Dominions Trust sought to gain publicity for its UDT Laystall racing team by breaking the Sydney-to-London flight record, engaging Flockhart for the attempt. He departed Sydney on 28 February 1961 in a Commonwealth Aircraft Corporation-built Mustang, registered G-ARKD, but the effort was abandoned at Athens due to engine problems after delays caused by bad weather. A second CAC Mustang, registered VH-UWB, was acquired in Australia for a renewed record bid. On 12 April 1962, while on a test flight in preparation for that attempt, Flockhart crashed VH-UWB in poor weather near Kallista, Victoria, and was killed.