Vemić turned professional in 1995. He reached a career-high singles ranking of world No. 146 on 25 February 2008 and a career-high doubles ranking of No. 31 on 12 January 2009. He is ranked 19th on the list of Fastest recorded tennis serves with a serve speed of 235 km/h (146 mph).
In 2007, Vemić began the season ranked No. 406, a drop of 170 places from early 2006. That year, he competed in Futures, the Challenger Tour, and selected ATP Tour events. His results included two Futures semifinals, one final, three ATP main draw qualifications, and a win in a top-level Challenger. These results returned him to the top 300 in April. He later reached the semifinals of the Challenger in Bermuda and qualified for the main draw of the French Open, where he lost in four sets to Janko Tipsarević. In August, he reached the semifinals in Segovia and the final in Binghamton, returning to the top 200 for the first time since 2004.
He won five doubles titles on the 2008 ATP Challenger Tour: in Miami, United States (with Ilija Bozoljac); Sunrise, Florida (with Janko Tipsarević); Cremona, Italy (with Eduardo Schwank); Waco, Texas (with Alex Bogomolov Jr.); and Calabasas, California (with Ilija Bozoljac).
In Grand Slam competition, Vemić reached the doubles semifinals of the 2008 French Open and the quarterfinals of the 2008 US Open, partnering with Bruno Soares.
Following his professional playing career, Vemić briefly coached WTA player Andrea Petkovic, assisting her and coach Petar Popović in reaching a career-high WTA ranking of No. 9.
Soon afterward, Vemić joined the coaching team of world No. 1 Novak Djokovic. As part of Djokovic's team, he coached alongside longtime head coach Marián Vajda during several major tournaments, including the US Open (2011, 2012), the Australian Open (2012, 2013), the French Open (2012, 2013), and Wimbledon (2012, 2013). During this period, Djokovic won two Grand Slam titles at the Australian Open and one at the US Open, as well as the 2012 ATP World Tour Finals in London.
In 2016, Vemić coached Djokovic at the ATP Miami Open, where Djokovic defeated Kei Nishikori 6–3, 6–3 in the final, surpassing Roger Federer in career prize money earnings.
In January 2013, Vemić joined the coaching staff of the Serbian Davis Cup team, helping lead the team to the Davis Cup final against the Czech Republic. He continued as part of the coaching staff during the 2016 Davis Cup World Group competition.
In August 2016, Vemić served as the official coach of the Serbian men's tennis team at the 2016 Summer Olympics in Rio de Janeiro. The team included world No. 1 Novak Djokovic, former world No. 1 doubles player Nenad Zimonjić, and top-30 ATP player Viktor Troicki.
From August 2016 to 2017, Vemić was the head coach of the American doubles team Bob and Mike Bryan, considered the most successful doubles team in tennis history.
From May 2020 to February 2022, he coached American player Brandon Nakashima.[4]
As of May 2025, Vemić shares the role of head coach for Novak Djokovic with fellow Serbian coach Boris Bošnjaković.[5]