Can a for-profit organization attempt to solve problems like rape, human trafficking, prostitution, weapons trafficking, money laundering?

Sure... why not? Thinking some variation of a private investigation firm. Finding clients who would pay to solve the problem might be difficult, but that's true in all businesses.

Blackwater and other mercenaries are doing this.

no

Some variation of Peter Thiel's Palantir Technologies maybe

Sure! It's called social entrepreneurship.