Posting to Drupal Form via External URL

I am writing an application (non-Drupal) that is POSTing to a 3rd party public form written in Drupal. The problem I am running into is that the 3rd party form never processes the request, it only returns the original form. Is there a trick to posting to Drupal forms? The URL that I am attempting to post to is:

https://www.modelaircraft.org/membership/verify?ama_number=123&last_name=smith&op=Submit&form_id=membership_verify_form

The response should have the following fragment:

<div role="alert">   <h2 class="visually-hidden">Error message</h2>   No match found for AMA 123 and last name smith. Please verify. 

But it does not.