Born in Burnley, goalkeeper Fulton has been with Liverpool since U-10 level. He was part of the first-team squad's 2015 pre-season tour to Bangkok, Brisbane, Adelaide and Malaysia, and with Danny Ward being loaned to Aberdeen for the new season he was the third choice keeper for the start of 2015/16. Fulton was sub keeper for the League cup 3rd round tie against Carlisle on 23 September 2015 and was on the bench for three more games in the first half of the season and penned a long term deal in December. When Danny Ward was recalled in January from his loan spell at Aberdeen Fulton was loaned to Portsmouth in League Two, where he made twelve league appearances and one in the FA Cup.

To further Fulton's experience he was loaned to Chesterfield for 2016/17, making 26 appearances in League One. At the end of the season he left Liverpool on a free transfer, joining Scottish Premier League side Hamilton Academical, where he was back up keeper for 2017/18, appearing six times in the league. He impressed enough to be given a contract extension until 2021. Again he was back--up in 2018/19, appearing just four times in the league, one of them at Celtic where he made several good saves but also made two errors gifting goals to the home side who won 3-0. A knee injury kept him out for the whole of 2019/20 campaign but the Accies remained faithful to him and he started 2020/21 as their first choice keeper. In 2022/23 he won the Scottish Challenge Trophy but also endured relegation to League One. A thigh injury had kept him out of the promotion/relegation play offs but he was back at full fitness for the start of 2023/24. 

Fulton regained his place as Hamilton's first choice keeper but he sustained an arm injury in December and was unable to regain it as the club pushed for promotion. As the season came to an end he confirmed he would be leaving on the expiry of his contract. He then signed a two year deal at Hearts to be a third choice back up, saying it was a great opportunity to learn from two international keepers. Fulton made three appearances for Hearts in the Premiership relegation phase in 2024/25.