The request to site.json is checking that an API request can be made with the values supplied for the API Key and Publishing Username that are set on WordPress. If you make the request with curl from your local computer, is it working for you? For example, this request with $api_key set to your API key:
curl -X GET 'https://forum.example.com/site.json?api_key=$api_key&api_username=system'
If this is woking for you, you could try installing the WordPress Health Check plugin and looking at the Secure communication row on the Site Status tab to see if it’s finding any issues with communicating with external services using secure connections (SSL.)