A 2-year-old girl was found in a nearby van critically wounded with gunshot in her chest.
A male family member who escaped the shooting was helping investigators, who refused to release any details but said they don't believe anyone else was in danger.
"What we have is a complicated death scene, and we're investigating all the possibilities," said Kevin St. John, a spokesman with the state Justice Department, which is leading the investigation.
Walworth County District Attorney Phillip Koss said the shooting was part of a domestic dispute, but he wouldn't elaborate until autopsies were completed and the crime scene was fully evaluated.
Police suggested earlier it could be a murder-suicide case.
"I don't believe they are actively looking for anybody right now," Koss said. "There are just a couple scenarios here we are not sure about yet."
Officers, responding to a report of shots fired, stormed the A-frame duplex Saturday night with weapons drawn, kicking in the door, neighbor Richard Heideman said. He saw two paramedics go in behind them and come back out minutes later.
"That's when I knew everybody was dead," Heideman said.
As the bodies were wheeled out, one onlooker dropped to his knees on a neighbor's lawn and threw his hands to the sky in prayer.
Authorities have not released the victims' names, but Kay Macara said her 19-year-old daughter, Vanessa Iverson, was among the dead.
With tears in her eyes, Macara said her daughter went to the apartment the previous night to visit friends.
"My child," she said. "I want answers."
Also in the house during the shooting were the twin infants, their parents, and the mother's sister and her husband, said Sarah Iverson, Iverson's sister-in-law.
The 2-year-old sister of the twins, Jasmine, was in serious condition at University of Wisconsin Hospital in Madison.
The father of the children, 22-year-old Ambrocio Analco, was among the dead, said Marco Pastrana, Analco's cousin. He said the twin boys were 2 to 3 months old.
Pastrana said Analco no longer lived with the children's mother. Analco left Pastrana's house Saturday night to drop the kids off with her at the duplex, Pastrana said.
"It came to my mind that this was not true," Pastrana said. "But when I came here I went to talk to the cops and they tell me that he was dead, his kids were dead and one was in the hospital."
Neighbor Leandra Mena, 65, said she heard what she thought were firecrackers coming from the house around 10:30 p.m. Saturday.
"I thought it was firecrackers because it's so close to the Fourth of July," she said.
Police cordoned off two blocks around the duplex for most of the day. On Sunday morning neighbors, some still bleary-eyed, clustered on the sidewalks, watching investigators move bodies out of the house.
Tina McKinnon, 37, lives about a block away and said there was never any commotion at the house. "The children were very pleasant," she said.
Delavan, home to about 8,000 people, lies in the farm fields and woods between Janesville and Milwaukee. The P.T. Barnum Circus, "The Greatest Show on Earth," was founded in Delavan in 1871, and statues of circus animals decorate the town square.