Leahy began his career with Rugby Town, affiliated with the Gordon Strachan Football Foundation. He signed for Falkirk in the summer of 2012, after trials with Birmingham City, Bradford City and Peterborough United. He was signed by Steven Pressley originally as an attacking midfielder, despite the fact he had played left-back for Rugby Town the previous season. He made his first team début in the Scottish Challenge Cup against Stirling Albion on 28 July 2012. His league debut came in September against Cowdenbeath at Central Park as a substitute in the 72nd minute of the game. On 10 November 2012, he scored his first goal for The Bairns against Raith Rovers at Stark's Park. However, he would only go on to make three further appearances that season; two of which were from the bench. Leahy continued to be a peripheral figure under Gary Holt as he struggled to stake a claim for a place in the side; on 7 September 2013, he was even listed as the substitute goalkeeper for Falkirk in their Scottish Challenge Cup Quarter Final against Raith Rovers, as Falkirk didn't have a third choice keeper to put on the bench. He made 23 appearances in all competitions in season 2013/14, most of which were from the bench, and managed a solitary goal; an injury time equaliser against Dundee at Dens Park. Peter Houston became Falkirk manager in the summer of 2014, just after Leahy had signed a new two-year contract extension. Houston was the first manager to give Leahy a chance to hold down a place in the side. He was on the bench for the first four league games of season 2014/15, but started the 1–1 draw with Queen of the South on 13 September 2014 at left-back, and has since made the position his own. Leahy made 41 appearances in all competitions; he also started the 2015 Scottish Cup Final for Falkirk against Inverness Caledonian Thistle, which they lost 2–1. He continued to be a mainstay in the Falkirk team that had a fantastic season finishing 2nd in the 2015/16 Scottish Championship, making 46 appearances across all competitions. On 22 August 2015, he scored what would go on to be voted Falkirk's goal of the season in a 2–1 win against Livingston at Almondvale. On 19 February 2016, Leahy penned a one-year extension with Falkirk, keeping him at the club until the summer of 2017. Leahy left the Bairns in May 2017, following the expiration of his contract. He was released by Walsall at the end of the 2018–19 season. On 9 July 2019, Leahy joined League One side Bristol Rovers.
