I don't think it is required, but it really makes people mad and will start to bad mouth you and your site if they can't find a way to contact you. Also I understand that it gives your internet (online or offline) business more credibility that you are a real person.
A good example is the other day I bought a course on how to use Pinterest as a business. It turns out that this course is three years old and that I had already watched all the videos in the course on Youtube. The training course offered a 100% money back guarantee, so I tried to contact the person for a refund or be transferred to a newer course if they had one.
There was to contact information anywhere for this person, no email, not P.O. box, not physical address. - Nothing.
So the next thing I did was to get on Facebook and told everyone about this course and person, not to be suckered in.
The bad news for this person is that about 4 weeks later a finally got an email from this person asking me how I liked that course. It turns out that this person did have an updated course and a facebook group plus an email contact at any time for any help. I'm now happy.
But the moral of the story is because they had no contact information in the course and on their website, I had no way to contact them and caused me to get on facebook to bad mouth them. - Not good for any business.
So I recommend you put some kind of contact information on or website, required or not.