Is there a whitepaper or article detailing the specific ports or protocols required to send the data using the Scanning Tool? When using the Web Client outside of our network, it doesn't recognize that its there and requests that it be installed again.

An example of the error we see in the Web Client is attached as well.

Web Tools Agent only uses https/443 to communicate with the remote web client server. What you see there is the html page running in the browser attempting to communicate with the Web Tools Agent running on the local machine. This is the mechanism it uses to e.g. launch the scanning application. Those are the 3 ports that it will attempt to use. It sounds like your machines are set to use different firewall rules when attached to different networks, and you'll need to update those. But note that this is kind of an unusual connection as it's between processes on the same machine - you don't need to allow remote requests on those ports, just local ones.