Shore played for the Archbishop Denis O'Connor (DOC) Chargers high school hockey team in Ajax, Ontario.

[2] While majoring in finance, Shore played 38 games for the Maine Black Bears during his freshman season.

[6] He was then signed to an amateur try-out contract with the Stars AHL affiliate in Texas to finish the season.

He got off to a quick start in his rookie campaign, as he was selected as the AHL's player of the month in October with 8 goals and 3 assists for 11 points with a plus 7 rating in 9 games.

[10] In his third full season with the Stars in 2018–19, having contributed with 17 points in 42 games, Shore was traded to the Anaheim Ducks in exchange for Andrew Cogliano on January 14, 2019.

Upon the conclusion of camp, Shore was signed to a one-year, two-way contract with the Oilers on January 14, 2021, and immediately placed on waivers.