This is an example of a OpenSearch API. You can use it for the search in your browser.
To test the changes locally, you need a secure URL pointing to our localhost environment (aka tunnelling). Feel free to use any tool you know, I can recommend ngrok or localtunnel. You can find other options on this blog post.
Visit the secure URL on your browser Refresh the page - the browser has read the link tag and indexed it Type the URL again and the search shortcut should be available
Firefox requires you to manually index the site as a search engine. The different ways to do it are described in their documentation.