Sissons played as inside left for Middlesex and England Schoolboys, but was moved to outside left by Hammers manager Ron Greenwood. He made his senior debut for the east London club on 4 May 1963, a home game against Blackburn Rovers. He became the youngest player to score in an Final at Wembley, in May 1964, and second youngest to appear behind his Preston North End counterpart Howard Kendall.
He went on to play 213 league (37 goals) and 52 cup (16 goals) games for West Ham.
Sissons moved to Wednesday in 1970 for £60,000. He made 115 league appearances for club, scoring 14 goals, before moving to Norwich, where he rejoined former team-mate John Bond.
Chelsea signed Sissons for £70,000 from Norwich City in August 1974. He made his debut on 17 August 1974 in a 0-2 home defeat by Carlisle United.
Making 11 appearances during the 1974-1975 season for Chelsea he failed to establish himself in a struggling Chelsea side who were relegated at the end of his one season resulting in Chelsea cancelling his contract.
He moved to Cape Town. Sissons finished his career at Cape Town City.