Use an actual form element for search box

[SoniEx2]

Summary

With an actual form element, search box can be added to browser. This is similar to #7058 in that it's about "add search to browser", but unlike #7058 this is done from the in-page search box, instead of somewhere in the browser's menus.

At least Firefox supports these 2 ways to add a search engine. So we should probably support both.

However, unlike #7058, this one is non-trivial. This would likely require a new widget, at the very least.

For example, GitHub provides both OpenSearch metadata (so the browser gives the option to add GitHub search), but it also provides a proper search form, so right-clicking the search box brings up "Add a keyword for this search". It's also worth noting that the search form does something slightly different from OpenSearch: the search form is local to a project, and the OpenSearch is global to all of GitHub!