Hello , thank you for the feedback.
Yes I tried to get the auto-focus on the search box when I initially created the add-on (the object is "autofocus" and in the code I force focus on it), but didn't succeed well as the sidebar simply does not seem to get focus at all when it is open, and doesn't seem to be able to get it by itself :-(
To see that, when opening the sidebar, simply do an action on keyboard, and you will see that keyboard keys go to the main web page only.
Autofocus by itself is working since when you click on a non selectable object of the sidebar (like the lifts), then you see that the search box gets the cursor .. so it is really a matter of forcing focus on the sidebar as a whole.

Now that I got much more experience on add-on development and sidebar programmation, I will try again (but no guarantee, it might be intentional by FF that a web extension [not native] sidebar cannot steal focus by itself).

Thanks, aaFn.

================== Edit =================
I didn't find any way to do that, and opened a bug to FF https://bugzilla.mozilla.org/show_bug.cgi?id=1502713 as this is not matching the normal behavior with native sidebars.