If you’re using the built in search (“sodo search”), the actual ‘search’ happens on your browser.
The js-script is loaded when you use the theme helper {{ghost_head}} in the default.hbs: <script defer="" src="https://cdn.jsdelivr.net/ghost/sodo-search@~1.5/umd/sodo-search.min.js" ....
It the same as the Ghost Portal: it’s an IFrame.
You can not directly apply styles to the inner IFrame element through parent window CSS class.
You can change it with another search tool “Algolia” - see the comments of @cathy_sarisky making-use-of-search
If have found a way to change the search-text - see:
https://forum.ghost.org/t/is-there-a-way-for-themes-to-call-a-function-when-sodo-search-is-activated/32812