Yes, there was an ongoing investigation by Chandler police into the killing of CV. It had not been completed when the whole issue about the children became foremost and the FBI took over to pursue that element.
It appears that as soon as CV was killed, some members of the extended family began to contact Chandler police alleging it was a planned conspiracy, and began feeding them emails, text messages, and the parts of the story they knew about.
Police investigations take time, especially when none of the players: AC, LV, CV, actually lived, or continued to live, in the community. All their previous history and ongoing activities were in other jurisdictions, so Chandler police would have had to get all the details from other police departments, particularly Gilbert police, try to confirm/piece together the story, before they could even begin to compile evidence of the conspiracy.
Arizona has a 'stand your ground' law, so the fact that CV did not live in the home, but Lori did, and her brother claimed to be protecting her against assault in her own home, IMO saved him from being arrested. I think LV/AC knew this, and scripted the killing to happen that way, luring CV to to home.
This link refers to the ongoing communication between Chandler police and extended family members.
Lori Vallow Daybell case: A breakdown of evidence in 2,500 emails released by Chandler Police | 12news.com