I found a 2001 article from PC Magazine, which said that the only time you see a number on the Amazon webpage is on the last screen you see when you have finalized an order.
I also found a small website called 'The Amazon.Com Customer Service Phone Number' page set up by a ticked off customer at:cliche-ideas
That is dated 3/2003.It shows an 800 number for customer service as:
1-800-201-7575. I saw that number in a few different webpages, although one said that it no longer worked. The webpage shows an international number too. [I think that just means that it isn't toll free] It also lists the corporate main number and FAX number.
Under the 'conditions of use' there is a phone number for their legal department; 206-266-4064. Maybe you could call that number and get a main number for them if you are dogged enough.
It sounds like they really really don't want to take phone orders.
If you want their number for some other reason than placing an order, maybe calling the legal department isn't a bad idea as a start. [I wouldn't give any credit information out on the phone unless you are certain the number is correct.]