How to Contact a Facebook Administrator
Facebook does not make it easy to contact its support staff. Leaving feedback or reporting a site issue gets your message to them, but the site clearly states that they may not respond. If you're trying to contact the administrator of an individual Facebook page, you may be able to message or post to the page, depending on the page's settings.
Report a Problem
From any Facebook page while logged in, click the upside down triangle representing the Options drop-down. Select "Report a Problem." For any concern or question that doesn't regard abusive content or a site error, choose "General Feedback" from the list of categories. Note that the feedback form says, "We don't typically respond to feedback emails, but we're reviewing them." If you select "Something Isn't Working," you'll see a screen to report errors. Describe the error and upload a screenshot of the issue if you've taken one. Choosing "Abusive Content" takes you to a Facebook help article on the topic with no actual contact information.
If you're trying to contact the administrator of a Facebook page, as opposed to someone who actually works for Facebook, do so directly on the relevant page. Click the "Message" button displayed on the page's cover photo. Facebook administrators have the option to disallow incoming messages, which is the case if the page has no Message button. If there is no message button, look for the text box to post on the page's Timeline. Administrators can also disable this feature, so you may not see it, either.