Blackman joined Chelsea at under-13 level and progressed through the club's youth system.[4] He was named by manager André Villas-Boas in a senior match day squad on 29 October 2011, when he sat on the bench for a Premier League match against Arsenal at Stamford Bridge.[5][6] He began to train regularly with the first-team squad in the 2011–12 season and was an ever-present as the under-18's won that season's FA Youth Cup with a 4–1 aggregate victory over Blackburn Rovers.[7] Blackman signed a new five-year contract with the club in June 2014 and extended that contract by two years in July 2017.[8][9] He left Chelsea at the expiry of his contract in June 2021, having never played a first-team match during his seven years as a senior professional.[10]
On 18 March 2016, Blackman joined Swedish side Östersund on loan until 31 May.[15] On 4 April, he made his Allsvenskan debut in a 1–1 away draw against Hammarby IF, playing the full 90 minutes.[16] He made twelve appearances, keeping three clean sheets, during his ten weeks at the Jämtkraft Arena as Graham Potter's Östersund finished in eighth-place.[5] On Sunday, May 22, 2016, after his penultimate match with the club ended with a 0-5 loss against GIF Sundsvall, he was stopped at four in the morning by the police and found to be drunk driving with 0.68 percent alcohol in his blood. Blackman played again the following weekend but his loan was not extended.[1]
On 15 August 2016, Blackman joined League Two side Wycombe Wanderers on loan until 3 January 2017. The following day, he made his debut against Accrington Stanley in a 1–1 draw at Adams Park.[17] On 20 August, Blackman kept his first clean-sheet for Wycombe in 0–0 draw at Blackpool.[18] He then conceded ten goals in the following three league games.[19][20][21] On 1 January, it was announced that Blackman's loan spell had been extended until the end of the 2016–17 season.[22] He made 52 appearances in all competitions for Wycombe as the club narrowly missed out on the play-off positions by just one point, finishing ninth. He was named as the club's Young Player of the Year and manager Gareth Ainsworth predicted that he would go on to play at the highest level.[23]
Loan to Sheffield United
On 27 July 2017, Blackman joined Championship side Sheffield United on a season-long loan. He was originally signed as second-choice for number one goalkeeper Simon Moore, however after an injury to Moore in pre-season, Blackman established himself as first-choice goalkeeper.[24] He made his debut on 5 August as the season began with a 1–0 win over Brentford at Bramall Lane.[25] However he was stretchered off after he picked an injury following a collision with team-mate Cameron Carter-Vickers during a 1–0 defeat at Queens Park Rangers on 31 October.[26] He returned to the starting line-up on 17 November, in a 3–1 win over Burton Albion.[27] Blackman lost his place in the side to Moore after picking up a groin injury in December, but returned in a 2–1 win over Yorkshire rivals Leeds United on 10 February, pulling off a vital save from Pontus Jansson.[28] On 17 March, Blackman received the man of the match award after a string of saves in a 0–0 draw with Nottingham Forest.[29] On 30 March, in a 1–1 draw at Brentford, he and opponent Ryan Woods were sent off for fighting over the ball after the hosts equalised.[30] With Blackman only being at the club on loan, manager Chris Wilder decided to play Moore in goal for the final five games of the season.[31] Blackman played 33 times in all competitions throughout the 2017–18 season, and United were believed to have been interested in a deal to bring him back for the following campaign.[32]
Loan to Leeds United
On 16 July 2018, Blackman joined Championship side Leeds United on a season-long loan to compete with Bailey Peacock-Farrell for the starting goalkeeper position.[33] He said that he was excited to work with manager Marcelo Bielsa.[34] On 14 August, Blackman made his debut at Elland Road, starting in a 2–1 win over Bolton Wanderers in the EFL Cup.[35] On 16 November, he suffered a broken tibia in his leg in a 3–0 defeat in an under-23 match against Birmingham City.[36][37] Due to the severity of the injury, he returned to Chelsea, leaving Leeds with just one senior goalkeeper in Peacock-Farrell.[38]
Loan to Vitesse
On 3 September 2019, Blackman joined Eredivisie side Vitesse on a season-long loan.[39] He reportedly was affected by injury and returned to Chelsea halfway through the 2019–20 campaign without having played a first-team game at the GelreDome.[40]