So I had another run-in with the frustrating Amazon captcha, appearing after I recently changed my amazon password.
Through troubleshooting, I logged into Amazon and went to edit my security settings, in which case I was asked to login TWICE IN A ROW on amazon to verify my identity. I had to enter the captcha (same image pattern as within the app), and then it asked for my signin and password again.
Long story short, I entered it, and got to my security settings. I then closed the window after satisfying amazon's (secret) double login test, and re-tried to sync my amazon account with Deliveries. I got one request for captcha and immediately asked what orders to load.
Posting here in hopes it helps others remove the week long delay and troubleshooting fun.