Exeter-born Joe Wragg had been with the academy for several years before he became an Exeter City scholar/trainee. His development and progress was, at times, hampered by injury before he made his only first-team appearance in an EFL Trophy tie in 2022.

Becoming a scholar/trainee in 2020 Joe made progress with the under-18s and, on 4 November 2020, he scored an extra-time winner at Home Park as Exeter City beat Plymouth Argyle 3-2 in the FA Youth Cup. He made further appearances in the competition in 2021/22. 

Joe made his first-team debut when he came on as a second-half substitute in the EFL Trophy at Forest Green Rovers on 18 October 2022. He was also on loan at Western League Wellington during the first half of 2022/23 and later, on a similar basis, moved to Southern League Bideford in January 2023. 

He was released at the end of the 2022/23 season and, after a spell with Weymouth's under-23s, signed for Wellington in February 2024.