As Chuck stated, logging in via the browser option when prompted may work for you, it didn't for me. What I recommend is quick easy and works everytime.
During Amazon's captcha process after successfully completing the challenge the next window to pop up is informing you that it is trying to text you a one time OTP code to verify your identity.
Well the vast majority of consumers usually don't take extra security precautions or even know what 2FA (two factor authentication) even is. So it's safe to say most won't already have this option enabled, making it impossible for the app to successfully send the OTP code required to complete the captcha.
Open the Amazon app on your phone > Click on the menu bottom in the top left > Your Account > Login&Security > Two Step Verification (2sv)
Click on the "edit" button and add your phone number. It will then send you a test OTP text to verify your new settings work.
Close out of both apps after saving those settings. Re-open the delivery app and log in either with the native app itself or the browser option. Now you can successfully complete both captcha challenges and for ever ending the evil loop.
Hope this helps.