Currently there is no way to email customer support. The options, as far as I can see, are live chat, or calling. Both require wait times. This means I, the customer, must stop what I am doing to wait for you. There are times when this might be okay.
However, email respects both of our schedules. If the issue is not urgent, I can send an email and don't have to put everything on hold to get a response.
Another reason to accept emails is that people may want to alert you to something, for example I was a potential victim of a phishing attack. I emailed Godaddy and got a reply saying "we don't do email" or something similar.
There are other ways I could report this phishing incident, I'm sure, but all of them were more time-consuming than an email, so I decided to just forget it.
So by not accepting emails you are potentially missing out on a lot of valuable intelligence. What is easy to share with you is more likely to be shared.
If you'd like to chat more about this I'd be happy to help.
How do I extract an old eMail from my server?
Christopher Carfi commented
Thanks! For members of the Pro program, email support *is* going to be supported. Scroll down to the "Need Help?" section and on the right-hand side of that section, there is an "Email us" link that will let you open up an email-based ticket.