Shellito spent his entire playing career at Chelsea, signing for the club from Chelsea Juniors at the age of 17 and making his professional debut two years later against Nottingham Forest. He featured in Tommy Docherty's re-built Chelsea side of the early 1960s as an attacking full-back, helping the club win promotion from the Second Division in 1962–63. However, as the side began to challenge for honors Shellito suffered a serious knee injury and despite several attempted comebacks, was forced to retire prematurely.
Upon his retirement, Shellito joined the Chelsea coaching staff and took charge of the club's youth academy in 1968. In the summer of 1977, he was appointed manager of Chelsea, succeeding his former partner at full-back, Eddie McCreadie. He managed to keep Chelsea in the First Division in 1977–78, but the highlight of his tenure was a shock 4–2 win over reigning European champions Liverpool in the FA Cup. He left the club less than a year later, with the side at the bottom of the First Division and facing relegation.