A hard-nosed right-winger, Mulvey played junior hockey with the Edmonton Oil Kings and the Portland Winter Hawks. He was selected in the 1978 NHL Draft by the Washington Capitals and played for parts of three seasons with the Capitals. Prior to the 1981–82 NHL season, he was sent to the Pittsburgh Penguins as compensation for the Capitals signing of Orest Kindrachuk. He would later be claimed on waivers by the Los Angeles Kings during the middle of the season.
It was during his brief tenure with the Kings that he would be involved in one of the most controversials incidents in the NHL. On January 24, 1982 in a game against the Vancouver Canucks, a fight broke out, and Kings' Head Coach Don Perry ordered Mulvey out onto the ice to fight. Mulvey, who had just returned from a recent suspension, refused, which angered Coach Perry who then accused him of not standing up for his teammates. Mulvey was benched for the rest of the game, and was placed on waivers a week later. Coach Perry would later be fined and suspended for the incident, but Mulvey would never play another NHL game, as the unfair perception around the league was that he was someone who would not stand up for his teammates.
After his playing career, Mulvey returned to the Washington, D.C.-area and settled in Reston, Virginia, where he bought a tennis club and turned it into a hockey facility with two rinks. His rink was instrumental in the growth of hockey in the Northern Virginia region and continues today under different ownership as Skate Quest of Reston.
For many years he was the head coach of the Reston Raiders of the Capital Beltway Hockey League. He then served as the head coach of the Virginia Statesmen of the Eastern Elite Amateur Hockey League and also coached Tier II hockey for the Prince William Panthers Hockey Club in Woodbridge Virginia.
His older brother, Grant Mulvey, had a long career with the NHL's Chicago Black Hawks.