This demonstrates how to use an SPFx Application Customiser to inject an additional search box into the header of each page, which uses the “Search Settings” (at either the Site Collection or Sub-Site level) to determine the redirect page.
Open this sample in the gallery (popup panel): Open in gallery
Source: https://github.com/pnp/sp-dev-fx-extensions/tree/main/samples/js-application-intranet-searchbox