Hello, @TFS,
In our theme, we have the Ajax Search element: https://postimg.cc/bZ4PX4LJ by using this you can add the search anywhere on your page as per your requirements and you can change its setting as well, but it will not exactly work as you want, it will show the complete site search result not the search result for the specific page only.
So you can also try to look for any third-party plugin as well, which better meets your requirements.
Regards 8Themes Team.