Google

The type of widget you're thinking of hasn't existed for a while. Google broke that functionality on Android a couple versions back. Some apps still make widgets that create the illusion it's working that way, but it really isn't.

So no, you can't have a functioning Firefox search bar on the home screen.

But you can make a widget that's basically a shortcut to the internal searchbar. One tap will:

Open the browser, create a new tab, place the cursor in the search bar, and open the keyboard.

It serves the same function: one tap then you can start typing your query and hit enter. It just doesn't do that on the homescreen.