Peter played college hockey at St. Lawrence University, leading the school to the 1988 NCAA Finals. His accolades during his collegiate career included being named ECAC Player of the Year and earned berths on the ECAC All-Star Team, NCAA All-American Team and NCAA All-Tournament Team.
He was drafted by the Calgary Flames in the supplemental draft but never made an appearance in a Flames jersey. Upon the end of the collegiate season in 1988 Lappin joined the Flames' IHL affiliate in Salt Lake City. He only appeared in the final 3 games of the regular season before he led Golden Eagles to a Turner Cup Championship with a playoff leading 16 goals and 28 points in 17 games. For his heroic performance he was rewarded by being named the IHL Playoff MVP.
Lappin went to the Flames training camp in 1988 but the 5'11" 180lb right wing from St. Charles, Illinois was buried int the Flames incredible depth. The Flames were already set at right wing with names like Hakan Loob, Joey Mullen, Lanny McDonald, and Tim Hunter. Over the course of the next seasons the Flames would add names like Sergei Makarov, Sergei Priahkin, Mark Hunter and Theo Fleury. This incredible list of wingers obviously meant that Lappin was well down on the priority list.
Lappin was sent back to Salt Lake in 1988-89 and lit up the league scoring 48 goals and 90 points in 81 games. The Flames traded Lappin to Minnesota for a draft pick in 1989. Although he would appear in 6 games with the North Stars, the bulk of his season was again played in the IHL, scoring 45 goals and 80 points with the Stars' affiliate in Kalamazoo.
After a poor training camp, Lappin would play the entire 1990-91 season in Kalamazoo, but sllipped to 20 goals and 67 points. The following summer Lappin became a member of the San Jose Sharks organization as the expansion team selected him from Minnesota in a Dispersal Draft. However Lappin appeared in only one game with the Sharks. He spent most of what to be proved to be his final season of hockey in the minors with the IHL's Kansas City Blades, scoring 28 goals and 58 points in 78 games.
Lappin collected no points in his 7 NHL appearances.