I was looking into how the pydata sphinx theme could support this search functionality, but was a bit confused by the documentation so asking questions here to help clarify.
The custom search input docs
Suggest that you need a structure like this:
<div role="search">
<form action="search.html" method="get">
<input type="text" name="q" placeholder="Search docs" />
</form>
</div>
However, in the pydata theme, we have a little button that will pop up a search window, rather than a search field:
Clicking this button, or typing Ctrl+K will cause the search window to pop up:

It has this structure:
<button class="btn btn-sm navbar-btn search-button search-button__button" title="" aria-label="Search" data-toggle="tooltip" data-original-title="Search">
<i class="fa-solid fa-magnifying-glass"></i>
</button>
Do we need to find some way to put an input in there for this to work? Or can we somehow get the sphinx RTD search to work with the button some other way?