Dominic Smith joined the Shrewsbury Town youth system at the age of seven, and was allocated a squad number by manager Graham Turner in December 2012, making the match-day squad for the first time in a League One home match against Brentford where he was an unused substitute in a goalless draw. He signed his first professional contract in February 2014, and spent most of the following season on loan at Conference North side Tamworth.

Smith made his Football League and Shrewsbury debut in a 4–2 home win over Colchester United on 10 October 2015, playing the full 90 minutes. Featuring regularly during the 2015–16 season, he signed a contract extension in March 2016, keeping him at the club until summer 2018.

Early the following season, Smith joined National League side Barrow on loan until January 2017, however he was recalled in October 2016 by new manager Paul Hurst due to an injury crisis at his parent club. Dom subsequently joined National League North side Southport on a half-season loan in July 2017.

On his return to his parent club, he was released by mutual consent in March 2018, going on to join local rivals AFC Telford United on a deal until the end of the season. On 11 January 2019, Smith was loaned out to Alfreton Town for one month. After playing five games for the club, it was announced on 13 February, that the loan deal had been extended until the end of the season before making the move permanent for the 2019/20 season.

Dom spent the next 3 seasons at Alfreton before joining the Robins for the the 2023/34 campaign.