In this problem of yours, a very easier way would be if there can be found some of their APIs or libraries that can let you do this.
But since you do not intend to use, or perhaps the service does not expose an API, the only way to do this is using an external library that will allow you to parse and validate the HTML response that you get back. For example,
HtmlAgilityPack[
^] is one of such libraries that I used to consume in my own apps back in the days. Use this library to work around with the HTML pages, and send new requests, get responses for the pages.
HttpClient Class (System.Net.Http) | Microsoft Docs[
^]
One problem with this is, that your HTTP clients might not be able to send the HTTP cookies, please see this link to find out how your cookies would be shared by a real browser,
How are cookies passed in the HTTP protocol? - Stack Overflow[
^]