Hi,
Well, you are running a live search there – even if there was a way to get those IDs, the search is not yet started during the page load. You could only fetch the post IDs via front-end javascript, then probably run an asynchronous request to send the IDs as query arguments, fetch the data, and replace the widget.
I am not sure if I recommend doing this, unless you have a very strong experience with both back-end and front-end coding, or have a good developer of doing that.