Born in Dublin, Fullam began his career at Home Farm before signing for Preston North End in 1958. He scored 6 goals in 49 league appearances for Preston. In 1961 he was asked to guest on Shamrock Rovers tour of America on the understanding that he would sign for the club
He scored his first goal in a competitive match from the penalty spot against Transport in a League of Ireland Shield game on 20 August 1961.
He was one of five Rovers" players who were on the League of Ireland XI side that beat the English Football League side 2–1 at Dalymount Park on 2 October 1963. Just over a week later he played against Valencia CF and the Rovers drew 2–2.
He scored against Real Zaragoza in the Inter-Cities Fairs Cup in 1965. He made 8 appearances in European competition for Bohs.
He was given a free transfer by Johnny Giles at the end of the 1978/79 season.
Fullam finished his League of Ireland career at Athlone Town, signing for them with Mick Smyth in August 1979. He played well over 400 League of Ireland games. He earned 11 caps playing for Republic of Ireland national football team, 10 while with Rovers.
He holds the Rovers record for the most European appearances with 19 scoring the aforementioned 3 goals.
, 1965, 1966, 1967, 1968, 1978, 1968/69, 1966/67.