Storey was born in West Bromwich and started his football education with Wolverhampton Wanderers but was released and whilst playing for Sedgley Lions under 15s in Division 3 – was spotted by Mick Elliott MBE who was the head of recruitment at Coventry City. He signed for Coventry City but after one full season was not offered a scholarship and was released. Again without a club, Elliott took Storey to Swindon Town for a trial game where he impressed the coaches and was handed a two-year scholarship. During his first season as a scholar at Swindon, he impressed and helped Paul Bodin's youth side to win their third successive season. After Swindon Town managerPaul Hart was dismissed as manager in April 2011, Bodin was installed as caretaker boss. Bodin brought Storey into the first team set-up for the club's home fixture against Oldham Athletic. He made his debut as a 78th minute substitute, replacing on-loan forward Calvin Andrew in a 0–2 loss. On 30 October 2012, Storey scored twice against Premier League side Aston Villa in the League Cup, to level the score at 2–2, before Christian Benteke scored a late winner to send Aston Villa through 3–2. On 6 May 2013, Storey missed a penalty in a shoot-out in the play-off semi-final against Brentford, as Swindon lost 5–4 on penalties, after the tie had finished 4–4 on aggregate. He was handed his first start on 10 August, and provided an assist for the game's winner against Stevenage. On 5 October 2013, it was announced that Storey had joined Conference Premier side Salisbury City on a month's loan deal. He made his debut on the same day in a 2–1 win against Wrexham. On 21 December, Storey returned to first team action for Swindon, and after coming off the bench in the 71st minute to replace Ryan Harley in a Football League One home clash against his former side Coventry City, he scored the winner 15 minutes later. On 14 February 2014, Storey joined fellow League One side Shrewsbury Town in a one-month loan. On 10 July, he moved to League Two side Portsmouth, also in a temporary deal. On 30 January 2015, Storey joined Newport County on an initial one-month loan. He made his debut for Newport on 31 January versus Shrewsbury Town as a second-half substitute. The loan was subsequently extended to the end of the 2014–15 season. He scored his first goal for Newport on 28 February 2015 in a 1–0 win against Burton Albion.
Inverness Caledonian Thistle (loan)
Storey joined Scottish Premiership club Inverness Caledonian Thistle during the 2015–16 summer transfer window on loan until January. He scored his first goal for the club on his debut against Heart of Midlothian in a 2–0 victory. In January 2016, his loan was extended until the end of the season.
Aberdeen
On 17 May 2016, Storey agreed to join Aberdeen when his contract with Swindon expired in the summer of 2016. He scored his first goal for Aberdeen on 20 August 2016, in a 2–1 win against Partick Thistle.
Partick Thistle
Storey moved to Partick Thistle for an undisclosed fee on 11 August 2017, signing a two-year contract. Storey scored his first goal for Partick thistle, scoring a 92nd-minute winner in a 2–1 home win over Dundee. In May 2018, it was apparent that his contract would not be renewed with the club. However, in July 2018, Alan Archibald extended Storey's contract with the Maryhill side with a revised one-year contract. Storey was released by the club in May 2019 after a disappointing season in the Championship.
For the start of the 2019/20 season, Storey rejoined Inverness on a permanent contract. Storey netted his first goal back at the club in a 2-1 win against Arbroath.
