Hello Heidi,
Thanks for asking. 
I can point you in right direction as this will require custom development and that falls outside the scope of support we offer. The closest I can think of to reduce your development work would be Icon 8 and Integrity 9 demo stack. So you can import any one of them, let’s say Integrity 9 demo to start the development work.
Next, to integrate search bar in between Logo and links you need to copy _navbar.php from /wp-content/themes/x-child/framework/legacy/cranium/headers/views/global/ to /wp-content/themes/x-child/framework/legacy/cranium/headers/views/global/. You may need to create child folders.
After that you can add following code to kind of output search form you are looking for. Ideally you should wrap the code inside a <div> element so that it would be easy for you to apply CSS.
<?php get_search_form(); ?>
Before making any changes, I suggest you to download and setup child theme from following resources.
https://theme.co/apex/child-themes
Believe I have shared all the necessary details that’s required to make necessary changes. Beyond this it will be difficult for us to provide you support on this topic. In case you are not comfortable with programming, I suggest you to seek professional help.
Thanks.