One of the people in the lobby, Pardeep Sidhu of Plainfield, Ind., said his aunt and uncle worked at the facility and that a friend's 45-year-old husband had been shot there.
Police say several of the injured are in hospital.
Ms Cook said four of the injured had been transported to hospital, one in a critical condition.
"All of us at Indiana University are appalled and devastated by the tragic and heart-wrenching events of last evening at the FedEx Ground facility in Indianapolis, and we mourn the victims".
FedEx released a statement early Friday saying it is cooperating with authorities and working to get more information. "My god. How many more days will we go to bed shook by the tragedy of another child killed by police, and wake up to news of lives lost in another mass shooting, before we do what needs to be done in this country to stop these things??" tweeted Maya Harris, vice-president Kamala Harris' sister and a prominent civil rights lawyer in her own right.
"I am communicating with local authorities to get all details of the attack and my office stands ready to help everyone affected any way we can", he said, later calling for change and "gun safety reforms now!" "And you know, Indianapolis Metro Police Department, the officers responded", Cook said.
FedEx said in a statement that cellphone access is limited to a small number of workers in the dock and package sorting areas to "support safety protocols and minimize potential distractions".
Some said employees were not allowed to have their phones with them while working shifts at the facility, making it hard to contact them.
It was at least the third mass shooting this year in Indianapolis alone. Less than a week later, 10 people were killed when a gunman opened fire at a grocery store in Boulder, Colo.
"Our most heartfelt sympathies are with all those affected by this senseless act of violence", the company wrote.
"Too many Americans are dying every single day from gun violence".
Chris Bavender, a spokesperson for the FBI's Indianapolis office, said that they are helping the police with the investigation.
A witness said he was working inside the building when he heard several gunshots in rapid succession.
"I saw a man with a submachine gun of some sort, an automatic rifle, and he was firing in the open".
Another witness told WTTV that his niece was sitting in her vehicle in the driver's seat when the gunfire erupted, and she was wounded. As officers arrived, they came upon an active shooter, she said.
Gov. Eric Holcomb ordered flags to be flown at half-staff until April 20, and he and others decried the shooting.
"She's fine, she's in the hospital now".