Born in Sheffield, South Yorkshire, O'Donnell started his career with Sheffield Wednesday's youth system. He was called up to first team at the start of the 2006–07 season and was an unused substitute for the League Cup match against Wrexham as back-up for Chris Adamson. O'Donnell joined Conference National club York City on 2 March 2007 on a work experience arrangement. He was recalled by Wednesday later that month, to provide cover injuries. O'Donnell signed his first professional contract with Wednesday in June 2007. He joined Buxton of the Northern Premier League Premier Division in December 2007 on a one-month loan, where he made two appearances. He joined League Two club Rotherham United on 8 January 2008 on a three-month loan, as back-up for Andy Warrington, but was recalled before the end of the deal as back-up to Wednesday's two first choice goalkeepers. O'Donnell was then loaned to League One club Oldham Athletic on 14 March 2008 on an emergency seven-day loan. He made his Football League debut the following day at the age of 19 in a match against Luton Town. He joined Conference Premier club Grimsby Town on 31 August 2010 on a one-month loan. He made his debut on 4 September 2010 against Luton, Grimsby winning 2–0 at home. On 4 February 2011, he joined Alfreton Town on a one-month loan. He made his debut for Wednesday in a 2–0 defeat away to Southampton on 19 March 2011. On 15 May 2012, O'Donnell was amongst the six first-team players to be released by Wednesday.
Chesterfield
Days after his release, O'Donnell signed for Chesterfield on 16 May 2012 on a one-year contract. After appearing as an unused substitute in the first five matches, he made his debut on 4 September 2012 as an 89th-minute substitute for Tommy Lee in a 2–1 home win over Oldham. O'Donnell made his first start four days later, in a 2–2 away draw against York on 8 September 2012. He appeared in the following 14 matches, mainly due to the absence of first-choice goalkeeper Tommy Lee, who was out for eight weeks. After Lee recovered and returned to the first team, O'Donnell was once again the club's back-up goalkeeper. On 14 January 2013, O'Donnell joined Conference Premier club Stockport County on a one-month loan. O'Donnell had his loan spell extended with the club until 16 April 2013. While at Stockport, he made 20 appearances and established himself as first choice goalkeeper, but the club was relegated to the Conference North. In the last match of the season, O'Donnell made his first Chesterfield appearance in over six months, a 4–0 home victory over Exeter City. He was released at the end of that one-year contract in April 2013 following limited first-team opportunities.
Walsall
Following his release from Chesterfield, O'Donnell signed a one-year contract with League One club Walsall on 18 July 2013. He was voted as their Player of the Year in 2014–15.
After losing his place to Jussi Jääskeläinen in the Wigan team, he joined Championship club Bristol City on 12 January 2016 on loan until the end of 2015–16 as cover for the injured Frank Fielding. He went straight into the team against Preston North End on the same day, in a 2–1 home defeat. After impressing head coach Lee Johnson, he signed permanently on 1 February 2016 for an undisclosed fee on a two-and-a-half-year contract.
Rotherham United
He re-signed for Championship club Rotherham United on 19 January 2017 on a two-and-a-half-year contract for an undisclosed fee.
O'Donnell signed for League One club Northampton Town on 8 January 2018 on a two-and-a-half-year contract on a for an undisclosed fee.
Bradford City
O'Donnell signed for League One club Bradford City on 27 June 2018 on a two-year contract, having been able to leave Northampton on a free transfer because of a relegation clause in his contract. He was offered a new contract by Bradford City in July 2020, following the expiry of his existing contract, signing a new two-year contract later that month.
