Starting from AmazonSync Prestashop module’s version 2.5.0 the synchronization method toward Amazon has changed. Modifications required by Prestashop and Amazon made it mandatory that your website always is reachable in order to downlad files.
If you are using external systems that might block requests toward your website (i.e. CloudFlare), make sure to add the following IP in whitelist:
51.210.100.177
The different requests all point to the following path:
https://www.mywebsite.com/module/amazonsync/getfile?list_of_parameters
Verify that all the requests sent toward links that contain module/amazonsync/getfile are accepted by your server to avoid missing any data.