He started his career in Alvesta GIF youth team in 1996, staying with the club until 2000 when then moved to Östers IF. He debuted for Öster in 2005, at the age of 17, in Superettan, helping the club reach Allsvenskan. The time in Allsvenskan was only 1 year and Öster returned to Superettan, however Robin missed large part of the season due to a cheekbone injury and saw the club get relegated again. Öster was promoted back to Superettan in 2009, however Robin decided to leave the club and signd for Halmstads BK, which were looking for a replacement for departing Magnus Bahne.
Malmkvist played the 2011 season on loan at Tromsø, as backup to Marcus Sahlman.
Malmkvist played Norwegian and European cup matches for Tromsø, before getting his league debut on 10 July coming on as a first-half substitute for an injured Sahlman away to Vålerenga, a match Tromsø lost 2-0. He have represented the Swedish U-21 national team in 2 games.