Science isn't really concerned with things that aren't wholly determined by physical causality. I'd say the scientific approach to free will would be neurological: e.g., what physical processes within the brain give rise to the emergence of free choice (be it an illusion or otherwise).

UPDATE: There are cases where a patient experiences loss of brain function leading to an inability to make decisions. They require a guardian to make even the most trivial choices for them (e.g. what to wear, in what order to eat the items in their meal, etc.) What exactly have they lost? An answer to this question, scientific or otherwise, would be relevant to free will.