Yes, i have the “SearchWP Live Ajax Search” plugin installed and it returns results without a submit button. In my searchform.php form i have an additional input like this:
<input type="hidden" name="post_type" value="book">
Where “book” is the slug of my Custom Post Type.
You can add a hidden input to your searchform.php (if you have one) to limit the results. Worked for me, without Relevanssi though.
<input type="hidden" name="post_type" value="your-cpt-slug">
Check here in the WP Code Reference
