Born in Enfield, London, Simpson was with Norwich City before he joined Arsenal at the age of nine, and played in an under-18s game at just 13 years of age. He featured prominently for Arsenal's reserve team whilst at Islington Arts and Media School, and was on the bench for the first team's League Cup match against West Bromwich Albion on 24 October 2006, but did not play. On 19 February 2007, he made history by becoming the first player ever to score a hat-trick at the Emirates Stadium, for Arsenal's under-18 side in an FA Youth Cup match against Cardiff City, in a 3–2 win.
On 31 August 2007 he was loaned from Arsenal to Millwall until the new year. The next day, he made his first team debut in a 2–1 League One defeat to Huddersfield Town at The New Den, replacing Danny Spiller for the final eight minutes. He scored his first goal in a 3–2 loss to Swansea City in the first round of the Southern Section of the Football League Trophy on 4 September, and his first league goal in a 2–1 defeat to Swindon Town on 29 September. His performance saw him earned November's PFA Fans Player of Month award for League One. On 15 December, he was sent off for celebrating in the crowd after equalising in a 2–3 home loss to Oldham Athletic. His loan was due to expire on 1 January 2008, but was later extended to the end of the season. Simpson scored eight goals in 44 appearances for Millwall and won the PFA Fans' League One player of the year award for 2008. Simpson thanked Millwall for giving him the chance to improve his game, stating "Going to Millwall was the best decision I have ever made. Playing regularly in the first team really developed my game. It helped me grow as a player and I made a lot of good friends. It did me the world of good. The Millwall fans were really good to me – and I appreciated their support."
He signed a new contract with Arsenal on 29 December 2008, but was then loaned out to West Bromwich Albion for the remainder of the 2008–09 season. He made his Albion debut on 3 January 2009, coming on as a half-time substitute for Robert Koren in a 1–1 draw with Peterborough United at The Hawthorns in the FA Cup third round. He scored his first goal for the team ten days later, in a 2–0 win in the replay. His first Premier League goal was scored on 31 January, in the 2–2 draw against Hull City at the KC Stadium. After making seventeen appearance and scoring two times in all competitions, the club were relegated back to the Championship after one season in the Premier League. Following this, Simpson returned to his parent club.
On 27 August 2009, Simpson joined the Championship's Queens Park Rangers on a season-long loan. He made his debut in a 1–0 away win at Scunthorpe United. His third game for QPR was against Cardiff City, at the Cardiff City Stadium, and he went on to score a first-half brace, followed up by scoring in the next game on 26 September 2009, in a 5–2 win against Barnsley. In three straight wins against Preston North End, Reading and Derby County between 17 October 2009 and 24 October 2009, Simpson scored three goals in three separate matches. As the 2009–10 season progressed, Simpson scored six more goals and despite struggling his goalscoring form later on, he finished as the club's top scorer this season with thirteen goals in all competitions.
On 19 August 2010, Simpson signed a three-year contract with Championship team Hull City for a fee of £1 million including add-ons. It came after when Simpson can leave Arsenal, prompting interests from clubs, such as, Queens Park Rangers, Fulham and Millwall.
In the 2011–12 season, Simpson made three appearances at the start of the season, but struggled to get first team football at the club. As a result, Simpson signed again for Millwall on loan, on transfer deadline day, 31 August 2011, remaining with them until the new year. He made his Millwall debut for the second time on 11 September 2011, where he played 72 minutes before being substituted, in a 3–0 loss against Birmingham City. It wasn't until on 18 October 2011 when he scored his first Millwall goal on his second loan spell, in a 1–1 draw against Brighton & Hove Albion. Two weeks later, on 29 October 2011, he scored twice, as well as, setting up two goals, in a 4–1 win over Ipswich Town. Simpson scored his fourth goal for the club on 20 November 2011, in a 2–1 loss against Bristol City. After making sixteen appearance and scoring four times in the first half of the season, Simpson returned to his parent club. However, Simpson spent the rest of the season out of the first team, as he remained on the sidelined.
At the end of the 2012–13 season, he was released despite keen on earning a new contract.
On 29 September 2013, he signed a two-year contract for Thailand's Buriram United which reportedly earned him £40,000 a month. On 1 February 2014 he scored his first goal in a Buriram United shirt during a 1–0 Kor Royal Cup win over Muangthong United in Suphanburi.
Simpson made his Leyton Orient debut, coming on as a second half substitute, in a 2–1 loss against Chesterfield. It wasn't until on 7 October 2014 when he scored his first goals in the second round of the Football League Trophy, in a 2–0 win over Dagenham & Redbridge. Four says later, on 11 October 2014, he scored his first league goal, in 2–2 draw against Sheffield United, followed up by scoring another two weeks later, in a 2–0 win over Doncaster Rovers. After scoring two goals throughout November, Simpson, however, was sidelined with injuries on two occasions that kept him out for the next three months. Simpson scored on his return from injury on 21 February 2015, in a 3–0 win over Oldham Athletic. Although he appeared out of first team around late-March and early-April, Simpson then returned to the first team and scored on 18 April 2015, in a 6–1 loss against Milton Keynes Dons. Although the club was relegated to League Two at the end of the season, Simpson finished his first season, making thirty-four appearances and scoring eight times in all competitions.
In the 2015–16 season, Simpson started the season well in League Two when he scored the opening game of the season, in a 2–0 win over Barnet. He then scored three goals in three consecutive matches throughout August against Stevenage, Newport County and Bristol Rovers. After scoring two times throughout September, Simpson, again, scored five goals in four matches, including a brace against Crawley Town on 10 October 2015. By the end of 2015, Simpson scored nine goals in ten matches against York City (twice), Plymouth Argyle, Wimbledon, Mansfield Town, Portsmouth, Bristol Rovers and Stevenage (twice). As a result of his performance, Simpson was named Sky Bet League Two Player of the Month for November. After scoring two goals by the end of January, Simpson went on a goal drought throughout February and didn't score again in a 2–2 draw against Carlisle United and scored a brace in a 3–2 win over Dagenham & Redbridge on 16 April 2016. A week later, however, Simpson was sidelined for a week, due to back injury. Up until this point, Simpson remained an ever-present player since the start of the season. At the end of the 2015–16 season, Simpson went on to make forty-eight appearances and scoring twenty-five goals in all competitions, finishing as the second's top-scorer behind Matty Taylor. For his performance, Simpson was named the fifth best player in League Two and Football League Two PFA Team of the Year. In addition, Simpson was named Away Player of the Year award at the 20th annual Supporters' Club Dinner-Dance.
Simpson became a starter from the first match against Vancouver Whitecaps and scored the Union's first goal of the season against 2016 Eastern Conference Champions, Toronto FC.
Simpson signed for Major League Soccer side Philadelphia Union on 9 January 2017 from English League 2 side Leyton Orient. Reports suggest that Simpson's departure was due to him being unsettled at the club. Upon joining the club, the club's Sporting Director Earnie Stewart, commented on his move, quoting: "Jay brings goal scoring prowess to our team, in addition to a valuable veteran presence, as he has demonstrated with multiple clubs in England."
