Was able to easily add my Ebay and Amazon accounts after I paid for the Permanent adfree version. But now a couple days later the Deliveries app removed my Amazon account. When I try to re-sync it keeps putting up the Captcha window and entering many codes never syncs my account just another Captcha window. I also reinstalled the app and without making any other changes to the Pro version the first thing I did was try to add my Amazon account and same result Captcha windows, but still added my Ebay account very easy.
After reading here I found I need to disable 2 step sign in verification also known as 2FA (2 factor authentication). I logged into my Amazon account, scroll to bottom of mobile site switch to full site and went to Change Account Settings. I found the 2 step verification and it has never been enabled on my account because I didn't even know about it. Another solution I might try is enable the 2FA on my Amazon account than disable it, reinstall the app and try again. But I don't want to do that first off because the app and sync worked great with 2FA having never been enabled on my Amazon account.
(When the problem first started I emailed the developer but haven't heard back so I came here. I'm going to send this post to the developer now that I have more info and wait for a response. Will update this as I learn more or if the issue is fixed.)
My phone received an update to the Deliveries app today 3.29.2016 5pm Pacific Time. I have a Note 3 with TMobile. First thing I tried after the update was adding my Amazon account and it worked great and no Captcha windows popped up. Had a delivery couple days ago from Amazon and a window popped up to add the delivery to my list.