Hello jde...
So, what you can do, in most cases is pass a thread with the URL.
For example, if the thread you have is onthe bottom of the page
www.example.com, you would do something like:
In a response you would get all the info for a thread in JSON, including the ID, which you can then use for your further calls.
If you cannot start from there, you can start by searching all the threads in your forum:
Your forum name is available in the Disqus dashboards.
You will get a list of threads with each one having its ID (and other info), so you can then use that data as future query params.
Depending on putting it to calls, this very much depends on what technology you use, but it generally works.
I have worked with C# HttpWebRequests, as well as with jQuery.ajax and $http AngularJS calls, they all work.
Hope I helped.
Mario