With CSS won’t be enough and you need to move your search code as well.
It should be something like:
<?php get_search_form(); ?>
and most likely you will find it in header-extensions.php file.
For more information you can use similar tutorial on how to add contact information. Just replace it with code from search form.
https://colorlibsupport.com/t/hot-to-add-text-under-social-media-icons-in-travelify-theme/#post-1553