A video on the petition's website features Vacca's four children, starting with his 12-year-old son, Christopher.
"It's legal for kids my age and younger to shoot Uzis," he said. "That hasn't changed."
Vacca's 16-year-old daughter, Elizabeth, offered a startling comparison: "Laws say that children can't drink, can't drive, can't vote. But they can shoot fully automatic assault weapons. That hasn't changed."
She further told CNN's "New Day" on Tuesday that their father often schooled them on gun safety when they were younger, telling them "how to be safe with guns, but he never let us fire them because we were too young."
It's unreasonable, she said, that children smaller than her little brother are able to handle automatic weapons "that military personnel are trained for weeks to handle."
"It's time for a change. We have a voice, and so do you," the children said on the petition's website.
"The adults haven't been able to keep people safe, so it's time for us to speak up," 15-year-old Tylor said.
On August 25, 2014, Vacca was teaching the 9-year-old girl how to shoot an Uzi at the Bullets and Burgers shooting range in Arizona. The gun range, which caters to Las Vegas tourists about an hour away, has said on its website that children between the ages of 8 and 17 can shoot if accompanied by a parent or guardian.
Cell phone video shows Vacca standing to the left of the girl as she fires. Gun experts have said that was the wrong position because the Uzi would have recoiled to the left
In the video, the girl fires several rounds in rapid succession. (Experts said an Uzi can fire five rounds in one-third of a second.) The gun kicks to the left as the girl loses control.
It's easy for anyone -- including adults -- to lose control of an automatic weapon. That's why beginners should have no more than three rounds in a magazine at a time, said Steven Howard, who runs American Firearms & Munitions Consulting.
"The thing begins to fire, and it begins to jump and buck all over the place," he said.
"Your first human instinct is for your hands to clamp down, and you clamp down on the trigger, and if the thing has a 32-round magazine ... it starts spraying all over and people get killed," Howard said.
Just days after Vacca was killed, his children said they forgave the girl who shot him. They released a video through their attorney, in which they read a letter they'd written to the girl they've never met.
"You are only 9 years old. We think about you. We are worried about you," Tylor said. "We pray for you, and we wish you peace. Our dad would want the same thing."
Christopher, the youngest, said he hopes to one day meet the girl and give her a hug. Elizabeth told CNN's "New Day" that it's important to her and her siblings' healing process that they get to know her one day.
"We hope we will get to meet her. It's a lot easier to not like someone when you don't know them," she said.
Added Christopher: "We didn't blame her. We never have."
And with the petition, they hope, no other families will have to suffer what these two families are enduring.