Norris began his career in the youth system at Hitchin Town. He made an appearance for Hitchin Town Arena in a 2–0 Hertfordshire Senior County League Cup defeat to Baldock Town Letchworth on 18 October 2008.
Norris joined League One side Brentford in the summer of 2010 on a two-year scholarship. Norris was awarded the number 35 shirt and received his maiden call into the senior squad for a league match versus Southampton on 30 April 2011. Beginning the match on the bench, he made his professional debut when he came on as a 75th-minute substitute for Jeffrey Schlupp during the 2–0 defeat. He was an unused substitute for the final game of the 2010–11 season, a 4–4 draw away at Huddersfield Town. Norris sustained an injury in pre-season which kept him out for the majority of the 2011–12 campaign, though he managed to score 4 goals in 7 Development Squad appearances and in April 2012 he made his second first team appearance in a 2–1 home defeat to Sheffield Wednesday, coming on as a substitute for Shaleum Logan after 81 minutes. That same month he was offered his first professional contract, a one-year deal to be part of the Development Squad. During the 2012/13 season, Norris failed to receive a call into the first team squad, but was Brentford's top scorer in the Professional Development League 2 South, netting 10 times in 18 appearances. He added a further three goals to his tally from as many Middlesex Senior Charity Cup appearances.
Norris signed a further one year Development Squad contract in June 2013. He made his third Brentford first team appearance on 27 August 2013, coming on after 61 minutes for Farid El Alagui in a 5–0 League Cup defeat away at Derby County. In early September, Norris crowned his first Brentford start with a goal, scoring in a 5–3 victory over AFC Wimbledon in the Johnstone's Paint Trophy. He made his first league appearance in eighteen months when he replaced Marcello Trotta after 70 minutes of a 2–1 away defeat to Stevenage on 12 October. On 2 May 2014, it was reported that Norris had been offered a new contract, after impressing in his loan spells away from Griffin Park during the 2013/14 season. Norris made three appearances and scored one goal for the Bees during the 2013/14 season, in addition to scoring one goal in five appearances for the Development Squad. Norris was tracked in late May by League Two clubs Stevenage, Dagenham & Redbridge and Northampton Town and it was reported on 21 June that he had 48 hours to decide his future. Norris left the club on 3 July, after turning down a new contract, later saying "after four years it was just time for me to move on. I have a lot to be thankful to Brentford for, as they took a risk on me at 16 and gave me a chance".
Norris joined Conference South side Boreham Wood on a one-month loan in December 2012 and made his debut in a 0–0 draw with Truro City on 8 December, replacing Cliff Akurang on 65 minutes. He made his first start in the following game, lasting 80 minutes of a 3–0 victory over Farnborough, before being brought off for Mario Noto. He found himself getting picked ahead of more experienced strikers Cliff Akurang and Ömer Rıza. Norris notched his first goal for the club with the opener in a 1–1 draw with Staines Town on 1 January 2013. On 4 January, Norris' loan was extended until the end of the month. Norris scored both goals in a 2–1 win over Maidenhead United on 9 January. The match proved to be Norris' swansong with the Wood, as he returned to Brentford shortly afterwards, having scored three goals in seven appearances. Looking back on his time with the club, Norris said "it's been very good, it's given me a lot more experience in the more physical side of football compared to where I am. It's nice to get some competitive games under my belt and hopefully I've done well here for the club".
On 18 October 2013, Norris moved to League Two side Northampton Town on loan until 16 November, linking up with fellow Brentford loanee Stuart Dallas. The following day he started in Northampton's match against Dagenham & Redbridge and scored the first league goal of his career in a 2–2 draw. He scored again in the following game, netting from 25 yards in a 3–2 away defeat to Rochdale and he bagged his third in three games with the equaliser in a 1–1 home draw with Cheltenham Town. Norris took his tally to four goals in five games when he scored Northampton's second in a 2–1 away win over Bishop's Stortford in a televised FA Cup first round match on 10 November. Norris continued his excellent goalscoring run and made it five goals in six games when he converted a game-winning injury time free kick against Fleetwood Town on 16 November. On 18 November, Norris' loan was extended until 4 January 2014. The return of Clive Platt and Izale McLeod to the team in late November saw Norris' goalscoring run end, as he lost his starting place and was relegated to the bench. Norris' final Cobblers appearance came with a start in a 0–0 draw with Portsmouth on 29 December. He was forced off after 34 minutes with an injury and returned to Brentford on 2 January 2014. During his time at Northampton, Norris made 12 appearances and scored five goals.
On 23 January 2014, Norris joined League Two side Dagenham & Redbridge on a one-month youth loan. He made his debut in a league match against Scunthorpe United on 25 January. Norris replaced Luke Howell just prior to half time and scored an 89th-minute equaliser as the Daggers battled back from 3–0 down to draw the match 3–3. In the following game, Norris made his first Dagenham start and scored the equaliser in a 1–1 draw with Southend United. Norris' loan was extended by a further month on 24 February. Norris bagged his third goal for the club with a "curled fine effort" in a 2–2 draw away to Exeter City on 8 March. A finish at the end of a jinking run saw Norris score his fourth goal for the club with the equaliser in a 1–1 draw with Bury on 22 March. On 24 March 2014, Norris extended his loan spell with the Daggers until the end of the 2013–14 season. Though he failed to score further goals, Norris made regular appearances through to the end of the season and played his final match for the club on 26 April, a 3–0 defeat to former side Northampton Town. Norris made 19 appearances and scored four goals for the Daggers. Upon returning to Griffin Park, Brentford academy director Ose Aibangee said "Luke has now proved he can score goals at League Two level and contribute in other ways".
Norris joined League One side Gillingham on 3 July 2014, with a compensation deal being agreed with Brentford six months later.
On 6 August 2016, Norris signed for League One side Swindon Town with a three-year contract for an undisclosed fee, later reported to be in the region of £150,000. He scored his first goal for Swindon in an EFL Trophy tie against Exeter City on 8 November 2016.As of match played 16 September 2017.