Born in IJsselstein to a Dutch mother and a Surinamese father, Vorm began his career at FC Utrecht. In order to gain more experience, he spent the 2005–06 season on loan to FC Den Bosch. He made his debut on 12 August 2005 against FC Emmen, the first of 35 appearances that season.

In August 2006, Vorm made his debut for Utrecht against Willem II. Although he joined FC Utrecht as the third-choice goalkeeper, behind Franck Grandel and Joost Terol, his pre-season performances impressed manager Foeke Booy, and saw him promoted to first choice keeper at the start of the league season. During this first season Vorm played 33 out of 34 matches, and his position as number one at the club remained unchallenged thereafter. Vorm missed a handful matches during the 2007–08 season due to knee and shoulder injuries. He left Utrecht at the start of the 2011–12 season, having played 136 matches in the Eredivisie.

In 2006, he was a member of the Netherlands squad that won the UEFA U-21 Championship 2006 in Portugal.

Vorm played his first match for the senior national team on 14 November 2008, replacing Henk Timmer at half time in a 3–1 home friendly win, conceding to Kim Källström within four minutes of coming on. He made his first start on 5 September 2009 in a friendly home match against Japan (3–0 win), followed up four days later by an impressive performance at Hampden Park during the World Cup qualifier Scotland – Netherlands, a match won 1–0 by the Netherlands. At Hampden Park, this was Vorm’s first competitive match ever outside the Netherlands.

Vorm was included in the preliminary squad for the 2010 FIFA World Cup in South Africa. On 27 May 2010, Netherlands manager Bert van Marwijk announced that the player would be part of the final squad of 23 participating in the FIFA World Cup. Before the start of the competition he was allowed to fly back to the Netherlands. During the World Cup, Vorm’s girlfriend Daisy delivered a son, Jaivy Waylon. The Dutch goalkeeper returned home for two days following his son’s birth.

In August 2011, Vorm joined Swansea City for a fee of around £1.5 million. He took the vacant number 1 shirt and made his debut for the club against Manchester City on 15 August 2011 in a 4–0 defeat. Although he conceded four goals, he also made eleven saves – more than any other Premier League goalkeeper, in a single match, in the whole of the 2011–12 season. In his second match for Swansea against Wigan Athletic, on 20 August, Vorm saved a 72nd-minute penalty from Ben Watson, to give Swansea their first ever point in the Premier League. On 6 November 2011, Vorm arguably saved Swansea from defeat against Liverpool and earned them the draw at Anfield by pulling off two outstanding saves to earn the clean sheet. On 10 December, he was instrumental in Swansea’s 2–0 win over Fulham as he saved Clint Dempsey’s 87th-minute penalty, with Swansea 1–0 up.

On 27 September 2012, Vorm signed a new four-year contract with Swansea. On 28 October 2012, he was ruled out for up to eight weeks due to a groin injury picked up in a match against Manchester City. On 3 November 2013, in the South Wales derby against Cardiff City, Vorm was shown a straight red card by referee Mike Dean for a stoppage-time foul on Fraizer Campbell. The match ended 1–0 to Cardiff City.

On 23 July 2014, Vorm announced via Dutch radio that he would sign for Tottenham Hotspur to compete with Hugo Lloris. He signed on a four-year deal costing Spurs £3.5 million after three years at Swansea, joining on the same day as Swansea’s Ben Davies made the same move and Gylfi Sigurðsson was exchanged as part of the deal. The transfer was disputed by Utrecht, who complained to FIFA that they were contractually due 30% of the fee received by Swansea. FIFA rejected Utrecht’s claim, and the case was heard in the Court of Arbitration for Sport in January 2016.

For the 2014 FIFA World Cup, coach Louis van Gaal included Vorm in the Netherlands squad, alongside young Ajax goalkeeper Jasper Cillessen who was first choice in goal and Tim Krul. Despite having a reputation for his penalty-saving abilities while at Swansea , Van Gaal opted to replace Cillessen with Krul in the penalty shootout during the quarter-final match against Costa Rica. Vorm came on for Cillessen in injury time of the 3–0 third place play-off win against Brazil in Brasília, thus ensuring that all 23 Dutch squad members played at the tournament.

Vorm’s first league appearance for Tottenham came on 21 March 2015, after Lloris went off injured in the fourth minute of a home match against Leicester City. Tottenham won 4–3. On 5 April he made his first league start, a goalless draw against Burnley at Turf Moor. On 13 August 2016, Vorm made his first appearance of the 2016–17 season for Tottenham Hotspur in a 1–1 draw with Everton at Goodison Park, in which he replaced the injured Lloris after 35 minutes.

Vorm’s contract was extended in 2018, however, in the 2018–19 season, he was replaced as the number two goalkeeper for Tottenham by Paulo Gazzaniga, having started in fewer games when Hugo Lloris was not available. After the 2018–19 season Vorm was released by Tottenham following the end of his contract with the club.

Three months into the 2019–20 season, Vorm re-signed on a contract until the end of the season as cover for Gazzaniga who had stepped into the starting eleven to replace the injured Lloris. Vorm made his first appearance of the season in the FA Cup match against Norwich City on 5 March 2020, which ended in a loss after a penalty shoot-out. On 27 July 2020, Tottenham confirmed that Vorm would be leaving the club following the end of the season.

On 26 October 2020, Vorm officially announced his retirement from professional football at the age of 37.