Jepson was born in Glasshoughton, near Castleford in West Yorkshire and after leaving school worked in the coalmines, playing football with Frickley Colliery F.C. in the Midland League. He was a late-comer to professional football and it was just after his 25th birthday that he was taken on by Huddersfield Town in May 1927. At that time Huddersfield were one of the top English football clubs, having won the Football League for three consecutive seasons in the mid-1920s, finishing as runners-up in 1926–27. With Scottish international Alex Jackson well-established at No. 7, Jepson had to content himself with a year in the reserves as Huddersfield again finished the season as League runners-up, coupled with a trip to Wembley for the 1928 FA Cup Final, where they were defeated 3–1 by Blackburn Rovers.
Southampton
In August 1928, Jepson was one of seven new signings as Southampton's manager Arthur Chadwick started to re-build a side that had finished a disappointing 17th in the Second Division in the previous season. Jepson went straight into the first team, making his debut in the opening match of the 1928–29 season, a 2–2 draw at Hull City. Jepson was injured in that match, and had to sit out the next two matches, before settling into the outside right position. In December 1928, he lost his place to Bobby Weale, who had just been signed from Swindon Town for a then club record of £1,000 Jepson regained his place at the end of March, playing in five of the last seven matches of the season, scoring three goals, as the "Saints" improved their league performance with a fourth-place finish. Weale was back in favour for the start of the 1929–30 season, but by mid-season his form had become erratic and Jepson was recalled to the side in March, playing in the last nine matches, scoring three goals which came in three successive matches in April. With Weale leaving The Dellin the summer of 1930, Jepson made the No. 7 shirt his own, missing only two matches during the 1930–31 season, in which the Saints finished in mid-table. Jepson again started the following season on the right, scoring in the opening match of the season, but an injury in November put him out for several weeks, before returning for a further four games in January, when he was again injured. During his absences from the side, manager George Kay tried seven different players on the right wing, before Dick Neal was signed in February. Neal's arrival brought Jepson's career with Southampton to an end, and although he was happy to remain with the club, he was put on the transfer list at a fee of £500. With no offers at this fee, he was given a free transfer to Fulham in June 1932 having made 95 appearances for the Saints, scoring 18 goals.
Later career
At Fulham, he found his favoured place at outside right occupied by another international, Welshman Billy Richards, and spent a frustrating season in the reserves. In May 1933, he returned to the south coast, joining Brighton & Hove Albion where he played two seasons in the Third Division South before retiring in 1935. After retiring from football, he became the manager of the Belgravia Dairy in Brighton. During World War II, he guested for Port Vale and Swansea Town, and in 1948, now aged 45, he was granted a special permit to join Sussex League side Hove as their player-coach.
