Let's put aside the natural/supernatural distinction and ask whether the scientific method could apply to ghosts, pixies, or elves.
The answer is sure: if a collection of phenomena is reproducible, we have something to study. We can then formulate testable hypotheses about why the phenomena occur. And, if we are able to conduct the tests, we can undergo the standard cycle to modify hypotheses to get progressively more accurate answers.
So, suppose that we can reliably record a ghost in a house, using cameras and microphones. We can use that data to study the phenomenon that can be given the name 'ghosts'. We can see when they appear, where the sounds come from, what the optical effects are, etc.
And I could go further. If the typical types of effects attributed to ghosts were verified, it would inspire a revolution in physics (at the very least). Sound waves obey certain laws under our understanding and if the moans and groan don't fit those laws, the laws themselves have to be re-investigated. The same goes for optical effects.
Now, because ghosts would almost certainly involve the violation of conservation of energy (and probably several other laws), we would want to be *very* sure about the effects observed, and those effects must be repeatable and very clear (not just on the margins of the error bars).
So, yes, if these were real phenomena, the scientific method would be able to study them.
But I will go further still: I predict, based on our understanding of physics, that no such phenomena will ever be seen in enough detail to allow verification of them.