Referring to site URL in shortcode argument

To answer the heart of your question: no, WordPress does not have a way of referencing posts by ID/slug/etc. within the post content. In fact, probably the recommended way to do that is with a…shortcode! Even that wouldn’t help you in this case, though, since you can’t nest shortcodes without a plugin.

So I’d recommend you use your own shortcode, or plug theirs if it’s a pluggable function. Simply replacing their URL argument with a get_permalink() that accepts an ID from the shortcode is probably all that’s required.