Edward Eveleigh was one of the club’s founding members. Having been an original St Sidwell’s United player in 1901, he appeared in the first game played by Exeter City in 1904 when he scored the team's opening goal with Sid Thomas getting the second. He briefly turned professional in 1908 although he did not appear for the first-team in the Southern League.

Playing for St Sidwell's United from the club's inception in 1901, he was recorded as a scorer in the opening friendly against Dawlish on 21 September 1901 and in the first league game Exeter YMCA on 5 October 1901. The club, which became Exeter City in 1904, had been formed largely by a group of 16- and 17-year olds who had attended St Sidwell’s school and lived close to the current St James’ Park.  

Having previously played for Exeter Wesleyan United, he appeared in nearly every St Sidwell’s game on record over the three seasons to 1904 and became a prolific goalscorer whilst helping Sid Thomas with some of his administrative duties. 

Moving into the Exeter City era, he scored the first goal in the opening league game against 110th Battery Royal Artillery on 10 September 1904 and appeared in nine of the first ten games in 1904/05 often captaining the team. Missing a February 1905 friendly against Torquay United, it was reported he had moved to Portsmouth and reports exist of a player of his surname having a trial with Portsmouth's reserve team in March 1905.

Known as 'Ginger' he was subsequently absent from City's line-up for all of 1905/06 but, with the club now featuring in the Plymouth and District League, reappeared towards the end of January 1907 to make seven appearances during the remainder of the season. Then, in 1907/08 he is recorded as playing in 25 games with his 15 goals making him the club's top-scorer.  

Exeter City: A Complete Record 1904-1990 records that, unlike most of City's other amateur players, he turned professional when the club joined the Southern League in 1908 although he did not appear in a first-team fixture thereafter.

[Like many players of the Edwardian era, there is uncertainty about some of the precise details relating to this player. The 1901 census for Exeter points to an Ernest Eveleigh - as opposed to Edward - a seventeen-year-old assurance clerk living in Albert Street. This would place Ernest as a contemporary of Sid Thomas and in the same line of work as Sid’s father. Although the ages are slightly awry there is an Exeter-born Ernest Eveleigh working in assurance in London in the 1911 census.]