Whether you're talking about Ancapistan or the real world today, if you find your property ransacked like that and don't have much in the way of hard evidence (high-res video), there's not much you can do. In both situations you'd have to do an investigation, look for clues, interview people in the area, stake out the local gangs, and maybe pay criminals to rat each other out. The only difference in Ancapistan is you'd be hiring private investigative agencies to do all that instead of depending on tax funded police.

The enormous cost of doing all that is one reason why I think you'd always have psuedo-government agencies that everyone in a community pays a regular fee to, like insurance.

What I'm not clear on is what kind of punishment would be carried out, assuming the persons you believe did the looting rationally refuse to cooperate with any judicial agency. Maybe that's when it's time to take the law into your own hands. If a court has publicly proven that they were the criminals, nobody in the community would fault you for exacting justice.