Some states (like Pennsylvania[1], for example) do have a process for "private criminal complaints". It's not without major drawbacks. The process can be (and has been) abused by people who simply want to make life difficult for others.

Texas also recently took the concept you've described but implemented the inverse with SB-8[2] which allows citizens to do precisely what the government itself is legally barred from doing.

[1]https://www.pacodeandbulletin.gov/Display/pacode?file=/secur...

[2]https://en.wikipedia.org/wiki/Texas_Heartbeat_Act