Sheehan joined the Swansea City Academy at the age of 10 and signed his first professional contract with the club in 2012. He was later loaned to Yeovil Town and Newport County, signing a permanent deal with the latter. In 2020, Sheehan made his senior international debut for Wales against the USA.

On 7th June 2021, Sheehan joined Bolton Wanderers and made his competitive debut for the club against Milton Keynes, which saw him score a stunning free kick.

During the summer of 2023, Sheehan signed renewed terms with Wanderers for a further two years.