Using the Chrome Dev Tools, you can record the network requests that occur using the Timeline tab.
It's actually using GET to their server and something unknown to Amazon (that would most likely be server to server and not accessible over Javascript).
Although the method on the form is POST, it has an onsubmit handler which calls the connect_amazon2() JS function (which makes a GET AJAX request on line 311) and then returns false, preventing the form from being submitted.
Yes, the Chrome Dev Tools are probably the easiest way to take a look at this type of information. I actually used Safari, but Chrome's tools are much easier to use.
It seems the form submit is being cancelled by the "return false" in the onsubmit, meaning that the form won't be posted normally and that method="post" irrelevant.
The connect_amazon2() must be making the GET request.
Also, would you say chrome developer tools is the lightest/most accessible way for the general public to inspect for such a thing?
EDIT: TY, schrodinger. That turned out to be a tab in Dev Tools that I just hadn't dug into before. No need to install anything!