Second time this has happened, and I finally figured out at least part of it. (Last time I just didn't buy what I was interested in.)
At the resolution I'm using (1920x1080) on Google Chrome on a 32" monitor, several windows critical to the checkout process - including entering new addresses for billing on a new card - don't scroll down far enough to see the buttons. Hitting enter on the form didn't autosubmit, either, and I couldn't see whether tab was getting me to the right spot.
I finally was able to make my purchase by zooming out to 75%, clicking the submit button, and then zooming back in, but not everyone will think of that; like I said, I didn't the first time.
Further, the checkout screen links you to saved addresses, but I didn't see a way to select or add one through that screen; I had to start the checkout process over, which was annoying.
Hey Derek, thank you for the feedback! Would you mind sending a screenshot or two of this to webmaster@paizo.com when it crops up? This would help us narrow down what's going on.
Next time I see it, I will!