I am not entirely sure what you mean. If you simply need to configure a URL as a POST, then you would build that URL in a string and the navigate your browser of choice to that URL. You could even use the static method from the Process object (Process.Start) to launch IE or Chrome and pass the POST URL as an argument.
Web page objects have a navigate method as well where you can provide a URL to navigate to (in this case your POST URL).
This is what I currently have. I have the key for the API listed in the authorization column. But when I click the button to run the API call and display results on a message dialog this pops up. It doesn't look like its able to find the API key that I put in. Also, where would I put the source for the API to look through. (This API takes in an HTML source and scans through it for accessibility errors)
Without access to your specific service it is hard to say. Perhaps attempting to call the service from a C# script might be better; particularly if you have access to Visual Studio to trace what is going on. It does look like you are doing things correctly though.