Lawyers for the family of 39-year-old Charles Vacca delivered a video message last week to the girl's family.
"You're only 9 years old. We think about you, we are worried about you, we pray for you and we wish you peace. Our dad would want the same thing," one of Vacca's four children said in the video, CBS affiliate KPHO reports.
"Our dad wasn't just an instructor. He was funny, strong, a protector," said one of Vacca's daughters on the video.
Said one of Vacca's sons: "Some day we hope that we can meet you, hug you and tell you that it's ok."
"It was important for Charlie's kids to share a bit about the father they knew and loved, and to express their feelings to the 9-year-old girl," attorney Marc Lamber said when he introduced the video statement. "Now we intend to work cooperatively with authorities and investigative agencies to determine how this could have happened and more importantly, how it could have been prevented. Ultimately, it is our hope that through this process, we will bring some semblance of closure to this family."
An attorney for the girl's parents has said earlier that the family is devastated by the tragedy that occurred Aug. 25 on a brief excursion during a vacation.
Recently released police reports named the child's parents as Alex Gen and Alison MacLachlan but don't list the couple's hometown.
New Jersey-based lawyer Kevin Walsh has said the family "prayed day and night that (Vacca) would survive his injury, and they continue to pray for his family during this terribly difficult time."
Vacca was shot at the Last Stop range in White Hills, Arizona, about 60 miles south of Las Vegas. He was flown to a Las Vegas hospital where he died hours later.
Prosecutors aren't filing charges in the case. Arizona's workplace safety agency is investigating the death.
The police reports said that immediately after the shooting, the girl said she felt the gun was too much for her and had hurt her shoulder.
Her family members were focused on the girl because they thought she was injured by the gun's recoil and didn't immediately realize that Vacca had been shot until one of his colleagues ran over to him, the reports state.
The shooting set off a powerful debate over youngsters and guns, with many people wondering what sort of parents would let a child handle a submachine gun. However, the reports don't explain why the parents let the girl take the Uzi.
James Goodnow, another attorney representing Vacca's family, said the family understands that the shooting has prompted a national discussion about gun safety.
"Although we feel this discourse is important and we intend to participate, Charlie's loved ones don't want it to overshadow the impact that this tragedy's had on everyone involved," Goodnow said.