Mkavu was shot in Uitenhage on Saturday night.
"No one has been arrested as of yet but investigations are ongoing," spokesperson Warrant Officer Basil Seekoei told Sapa on Tuesday.
The councillor was killed while he was inside his car. Seekoei said the gunman is believed to have approached Mkavu’s vehicle from the side and shot him through an open side window.
"He was hit in the upper body," he said.
Seekoei could not comment on whether the killing was politically motivated but said nothing was taken from Mkavu.
"I cannot speculate on the motive but investigations are at a sensitive stage at the moment," he said.
Mkavu was an ANC councillor.
The party's provincial spokesperson Mlibo Qoboshiyane called on law enforcers to speedily apprehend those responsible for the councillor's death.
"There is no justifiable reason that could inspire killing of any person, torching of a person’s property as was done to... Mkavu. With heavy hearts, we urge the SA Police Service to beef up its investigations so that they find those behind this evil murder," Qoboshiyane said in a statement.
Mkavu's house was burnt six months ago by angry residents of Uitenhage who were protesting.
Qoboshiyane called on other councillors to emulate Mkavu's example of standing by decisions taken by government structures.