If I understand correctly what you’re trying to accomplish, you’ll probably want “post meta” (post meta data) to be updated in relation to the 3rd party program, then adjust the post query to look for it before outputting posts.
This objective can be accomplished in different ways, the simplest ones not necessarily requiring input from or interaction with the 3rd party program: For instance, when a particular button is clicked, the post meta could be added, with all operations taking place within your installation.
It’s also very possible, however, that the implementation is already adding post meta data, in which case all you’ll need do is identify it. You might try outputting the results of get_post_meta() for a particular likely post, and seeing if there are any keys that look like what you want.
Either way, you would then use a query argument looking for the post meta value (aka custom field value) to determine which posts are output using WP Query. You can find details and some simple examples in the WordPress Codex here: https://codex.wordpress.org/Class_Reference/WP_Query#Custom_Field_Parameters