Search button or search field in header

The simplest — but perhaps least ideal — solution to this would be to create a new “Search” page, and place a Search block inside of it. Then, you’d just need to include that page in the menu. You could even style it with CSS to show up as a search icon.

To get a full search form, the process outlined in this post seems to work, though it seems like a bit of a hack to me:

https://divinotes.com/add-the-wordpress-search-field-into-your-primary-bar/

A better approach would be to modify the theme’s template-parts/header/site-nav.php file to include get_search_form() somewhere, and then to add styles to support it.