I'm pretty sure I've stumbled on something heavily unethical going on on the site, and I'd like to present my evidence to a moderator.
How can I get in contact with a moderator?
Or can I get a moderator to contact me?
You can talk to us via chat, but that is visible to everybody. I think this is by design, so that everything is "in the open". So to your general point, I don't think there is a way to do that through the StackExchange system.
Moderators can contact you, but that's for violations. The main way for users to draw the moderators' attention to things is by flagging.