I'm trying to make a Web Service Connection to a WSDL and getting the follow

Gene, please try the following.
1. Try accessing the service from the Appian server (irrespective of if the Appian is in the same network or otherwise). Open a browser on the Appian server and try the URL.
2. Check if there are any Firewall rules set up (This may not be an issue with WS, but please check either-way)
3. If the URL is reachable, use the getHttpPost() function from Send-HTTP smart service (configure it within a script task of a basic process model) and try executing the same.

As Brett says, 404 indicates that the resource is not reachable. You need to identify why. Eliminating the various options would be the first step.

Gene, please try the following.
1. Try accessing the service from the Appian server (irrespective of if the Appian is in the same network or otherwise). Open a browser on the Appian server and try the URL.
2. Check if there are any Firewall rules set up (This may not be an issue with WS, but please check either-way)
3. If the URL is reachable, use the getHttpPost() function from Send-HTTP smart service (configure it within a script task of a basic process model) and try executing the same.

As Brett says, 404 indicates that the resource is not reachable. You need to identify why. Eliminating the various options would be the first step.