Reading this just made me decide to remove the Google search bar from my home screen and put the Firefox widget there instead, but I'm a little disappointed that it really just functions as another button to open Firefox instead of letting me use the bar like it visually appears it should be.
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.
Yeah, nova launcher is the broken record response but there really is a reason for it. Its the stock launcher on crack, use it out of the box or spend hours tuning it into your own thing, while being super easy to use either way.