Chuma died on Tuesday in hospital in the northwestern Kenyan town of Kericho after collapsing at his home, his family said.
"Chuma was one of the foremost athletes to put Kenya on the world map. It's indeed a sad day for both Kenya and the entire athletics fraternity, He was an exceptional man," Athletics Kenya chief Jackson Tuwei told AFP.
"Kiprugut's medal from the 1964 Olympics in Tokyo is an extremely precious medal as it was Kenya's first ever at the Olympic Games and inspired more athletes who were to follow to shine on the Olympic stage," the federation said in a statement.
Chuma won the bronze in the 800m to make history in Kenya's Olympic debut in Tokyo. He was beaten by the legendary New Zealander Peter Snell who won gold, and second-placed Canadian William Crothers.
At the 1968 Olympics in Mexico he took silver in the 800m, passed down the straight by Australia's Ralph Doubell who went on to set a then world record of 1min 44.3sec.
After his retirement from the track, Chuma was employed as a field supervisor in tea plantations in Kericho owned by British multinational Unilever.
Athletics Kenya (AK) hailed Chuma, a former Kenyan army officer, as an "inspiration" to the next generation of the country's legendary track and field stars.