Generally, VPNs and Chromecast work together without issue. However, there are a few things to keep in mind.
First, make sure that your VPN is set up to work with your Chromecast. Some VPN providers may not be compatible with Chromecast, so you'll need to check with your provider to make sure.
Second, keep in mind that your Amazon Prime Video account may only work in certain countries. If you're trying to use your Amazon Prime Video account while connected to a VPN server in a different country, you may not be able to access your account.
Finally, if you're having trouble connecting your Chromecast to your VPN, you may need to disable IPv6 on your Chromecast. You can usually do this in the settings menu of your Chromecast.
If you follow these steps, you should be able to use your VPN and Chromecast together without any problems.