Aaron started his career in the youth system at Heart of Midlothian, although he had a spell with the academy teams at Celtic. In 2019, he became the youngest player since 2008 to appear in the Scottish Cup final when he played for Hearts against Celtic. He scored his first senior goal in the Edinburgh derby when Hearts won 2-1 away at Hibernian in September 2019.

In 2020, Aaron completed a move to Italian side Bologna. He made his first Serie A appearance on 28th September in a 4-1 win against Parma. He was also nominated for the 2021 Golden Boy award.

In the 2021/22 season, he scored his first goal for Bologna, in a 2-2 draw with Genoa. On the 24th April 2022, Aaron became the first Scot to score 5 goals in a single Serie A season since Graeme Souness.

Aaron signed for Brentford on a four year contract in July 2022.

Aaron has represented Scotland at many ages, and started his international career with the U17 squad in 2019. He has progressed to the senior international team, making his debut for the senior team in March 2022 in a 1-1 draw with Poland.