On December 13, 2012, it was announced that Trapp had signed with his hometown Major League Soccer club, Columbus Crew, as a Homegrown Player. He became the fourth HGP signing in club history, following in the path of Aaron Horton, Matt Lampson, and Ben Speas. After missing the early part of his first season while on international duty, Trapp made his professional debut for the Crew on July 7, 2013, starting and playing the full ninety minutes in a 1–0 victory against Portland Timbers; he was named as the club's man of the match after completing 86 percent of his passes. Trapp started the final sixteen matches of the season for Columbus, earning a postseason nod as the club's Breakout Performer of the Year.
Trapp received his first call-up to the U20 national team in January 2012, part of a training camp held in Florida. As the camp did not include a match, Trapp had to wait until June 6 to make his official debut for the team, starting a 4–2 defeat against Uruguay at Estadio Gran Parque Central. Trapp was named in the American squads for the 2013 Toulon Tournament, the 2013 CONCACAF U-20 Championship, and the 2013 FIFA U-20 World Cup. He scored his only goal at U20 level on February 26, 2013, providing the goal to send the U.S. to the U20 World Cup at the expense of Canada in the CONCACAF Championship quarterfinals, and was selected to the tournament's Best XI. Trapp capped his time with the U20s at the World Cup, playing all 270 minutes and captaining the side against France, although the United States failed to progress out of the group stage. He scored one goal in 21 appearances at U20 level, and was named as the U.S. Soccer Young Male Athlete of the Year in 2013.