A QPR youth, Goddard was West Ham's record signing when he moved to Upton Park for £800,000 in 1980. He was West Ham's top scorer in the 1982–83 season, and scored 71 goals in 213 League and Cup matches for the club. One of these goals was in the replay of the 1981 Football League Cup Final defeat to Liverpool. At the start of the 1985-86 season he partnered Tony Cottee in the West Ham front line, but injury ruled him out after one game. This led to Cottee being partnered by Frank MacAvennie, and the MacAvennie-Cottee partnership yielded 54 goals in one season, taking West Ham to 3rd place in the league.