Is it an HTTPS site? Are you configured to use only TLS 1.2? If so then SSIS won't work properly without a registry change to have its .NET 4 runtime enable the newer TLS protocol as documented here.
What happens when you use the connection manager to test the call?
If not done already then create a script task that uses the connection and calls AcquireConnection. Wrap that in a try-catch and capture the actual error coming out.
Also make sure that any firewall rules allows communication with the new web server you set up.