2009-10: Joe Wegwerth played for the Westchester Express bantam team in New York. He was selected for USA Hockey's Select 14 Player Development Camp.
2010-11: Wegwerth appeared in eight games for the Brewster Bulldogs in the Empire Junior B Hockey League as a 14-year-old and was Brewster High School's second-leading scorer as a freshman. He scored 1 goal with 1 assist and had 6 penalty minutes with the Bulldogs. In 16 high school games he scored 28 goals with 27 assists. Wegwerth was invited to USA Hockey's Select 15 Camp.
2011-12: Wegwerth was a team captain for the Brewster Bulldogs in his first full Empire league season. He scored 14 goals with 32 assists and had 52 penalty minutes in 32 games. Wegwerth scored 2 goals with 2 assists and had 18 penalty minutes in six games for the USA U16 team at the Winter Youth Olympic Games. The USA lost 7-5 to Canada in the bronze medal game. Wegwerth was selected by Belleville in the 13th round (252nd overall) of the 2012 OHL Draft. He attended USA Hockey's Select 16 camp and was invited to join the USA National Team Development Program in Ann Arbor in 2012-13.
2012-13: Wegwerth skated for the NTDP's U17 team in his first season in Ann Arbor; missing several games due to injury. In 27 regular season games he scored 7 goals with 6 assists and had 34 penalty minutes. In four games at the 2012 Four Nations Tournament in Switzerland he scored 3 goals with 1 assist for the first place USA squad. Wegwerth played four games and had one assist as the USA finished third in the 2013 Five Nations Tournament in Finland. In December he committed to playing college hockey at Notre Dame in 2014-15.
2013-14: Wegwerth returned to Ann Arbor for his second season with the NTDP; skating for the program's U18 team. In 46 regular season games he scored 3 goals with 3 assists and had a team-high 99 penalty minutes. He skated in the USA Hockey/CCM All-American Prospects Game in September and was ranked 141st amongst North American skaters in Central Scouting's mid-term rankings prior to the 2014 NHL Draft.