I am trying to figure out how to grab the contents of the XUL ‘searchbar’ element in such a way so that it is compatible with the current WebExtension api. Alternatively, I would like to see the ability to add arbitrary textboxes and other html elements to the toolbar (I understand that there was some talk of doing this, but I don’t know whether it has been implemented yet). I (and I think I am not alone) find it very frustrating that the old addon frameworks have been depreciated before full functionality is in place to replace the old code.
(Note, I posted a similar question a while back and I was just wondering whether there has been any progress)
The current list of available APIs can always be found here: https://developer.mozilla.org/en-US/Add-ons/WebExtensions
To answer your question: no, toolbar additions are limited to a button.
You can add search providers to the search box from a WebExtension add-on, if that helps.