Oh, Happy Day!!! 
I was able to order an Echo about 15 minutes ago for $99 (Prime member price)! It's scheduled for delivery on Friday, January 2. I put in my invite request on November 7, so it's been a long wait, as I know it's been for many others here.
It's been reported in the past that customers were able to purchase before actually receiving an invitation (it was probably scheduled to be sent soon). I haven't yet received an invite but was able to add an Echo to my cart and complete checkout with this link:
http://www.amazon.com/gp/offer-listing/B00L9EOQCO. Try it now, even if you've tried before!
I'm keeping fingers crossed for everyone who wants one!