The students are demanding that those who owe the university up to 150,000 rand ($A12,665) in fees still be allowed to register for the new academic year.
The cost of university education, prohibitive for many black students, has become a symbol of the inequalities that endure in South Africa more than two decades after the end of apartheid.
Wits student representative council president Mpendulo Mfeka said the person shot was not believed to be a student.
The Independent Police Investigative Directorate (IPID) said it was looking into the incident.
“The Directorate has dispatched four investigation officers to probe the allegations, the team will also cordon the crime scene and conduct the investigation including locating the person with the video footage, identify eye witnesses etc,” it said in a statement.
“IPID will ensure the body is protected and carried safe to the mortuary.”
Most students in South Africa are caught up in a group widely called the ‘missing middle’, those who are not poor enough to qualify for government financial aid but not wealthy enough to pay.
Protesting students took to the streets of Braamfontein district, where the university is located.
“The police threw stun grenades to try and disperse the protest but we didn’t run away, so they started firing rubber bullets, the person who was shot was shot directly in the head,” said Mfeka. “
A Wits spokeswoman said the university would issue a statement later.
In 2016, widespread student protests for free higher education forced temporary closures at a number of South African universities.