Oxley joined Rotherham United as a youngster and was promoted to the club's first team during the 2007–08 season. After being scouted by several clubs, he joined Hull City in August 2008 for a fee of £150,000. On 25 February 2010, Oxley signed a one-month loan deal with Grimsby Town, becoming the third Hull player to be loaned to Grimsby that season. Both Hull and Grimsby had suffered relegations by the end of the 2009–10 season. On 26 October 2012, Oxley went on a month-long loan spell to Burton Albion, but had to return to Hull following a broken finger after some good performances. Oxley signed a new two-year contract with the Tigers on 11 June 2013 following the club's promotion to the Premier League. In July 2013, Oldham Athletic signed Oxley on a season long loan, but he was recalled in March 2014. In July 2014, Hibernian signed Oxley on a six-month loan, with a view to extending to the end of the season. In June 2015, he signed a two-year contract with Hibernian and was the first choice goalkeeper for most of the 2015–16 season. Oxley signed a two-year contract with Southend United in July 2016. After a string of stunning performances during the 2017–18 season, Oxley was voted fans Player of the Season and Players' Player of the Season at the end of the campaign. On 4 June 2018 Oxley signed a new three-year contract at Southend and in August 2018, was named as captain. On 1 July 2021, Oxley joined Harrogate Town following Southend's relegation out of the Football League.