Fitzsimons began his career at Crystal Palace. On 17 January 2013, he joined Isthmian League Premier side Harrow Borough on an initial one-month loan to replace the departed Elvijs Putnins. He returned to the Isthmian League Premier on a one-month loan at Hendon in December 2013. He made his debut in a 6–0 win at Cray Wanderers on 17 December. He went on to play a total of eight league games and one cup game for the "Greens". On 10 September 2013, he joined Conference South club Havant & Waterlooville on a one-month loan to provide cover for the injured Scott Bevan. He made his debut the "Hawks" he following day in a 2–1 win over Farnborough at West Leigh Park. He made three further appearances for the club before returning to Selhurst Park. On 20 March 2014, he joined Farnborough on a one-month loan, and made his debut two days later in a 3–1 victory at Bishop's Stortford. He played a total of ten games for the "Yellows" before returning to the "Eagles". He was released by Crystal Palace in May 2014. On 1 July 2014, Fitzsimons signed a two-year contract with Championship club Bolton Wanderers. On 16 December 2015, he joined National League South club Bishop's Stortford on a one-month loan. The loan was later extended into a second month, and he made a total of eight appearances for the "Blues". He was released by the "Trotters" in May 2016. On 16 August 2016, he joined National League club Braintree Town on a short-term deal to stand in for the suspended Jamie Butler. He made his debut for the "Iron" later that day in a 2–1 defeat at Maidstone United. He joined Chelmsford City on 25 August, and manager Rod Stringer released goalkeepers Joe Welch and Colan Balkwell in the week following Fitzsimons's arrival. On 28 March 2017, he kept a clean sheet as Chelmsford beat East Thurrock United 1–0 in the final of the Essex Senior Cup. He made a total of 40 league appearances for the "Clarets", helping the club to qualify for the play-offs with a fourth-place finish; the club reached the play-off final, but were beaten 2–1 by Ebbsfleet United at Stonebridge Road.
On 27 July 2017, Fitzsimons signed with Notts County following a trial period. Manager Kevin Nolan said that he expected Fitzsimons to challenge Adam Collin for a first team place. Fitzsimons stated that he was looking forwards to working with goalkeeping coach Mark Crossley, who he said "has a very effective way of coaching". He made his debut on 8 August, in a 3–3 draw with Scunthorpe United at Glanford Park in the first round of the EFL Cup; County lost the resulting penalty shoot-out 6–5. He was offered new contract by Notts County at the end of the 2017–18 season.
On 16 January 2020, Fitzsimons joined Chesterfield on loan for the remainder or the 2019–20 season. He was recalled by Notts County on 24 January following an injury to Sam Slocombe.
