Stander, 25, narrowly missed out on a medal at the London Olympics, coming in fifth in the men's cross country event. He came in 15th at the Beijing Games and has been one of the top cyclists on the World Cup circuit over the past few years.
Police said Stander was killed on Thursday while cycling alone. He died at the scene of the accident in Shelly Beach, about 120 kms southwest of Durban.
"The taxi driver allegedly stopped after the accident until the arrival of the police. No arrest has been made at this stage," police said in a statement.
Stander was one of the biggest stars in South Africa's cycling community and his death touched off a flood of comments on social media.
"This is a massive tragedy and we are all stunned at the suddenness and impact that this will have on everybody in the world of cycling," KZN Mountain Biking, which represents Stander's home province, wrote on its Facebook page.