Can Anakin/Darth Vader use the Force with his metal arm?
Like I said, questions about the physical limitations of metaphysical abilities always lead to trouble, but here it goes....
When Vader raises his arm to telekinetically strangle someone, he is raising his mechanical replacement limb. With that particular ability, though, the raising of a limb is not actually part of the power. The arm is not acting as a conduit for the Force. Raising an arm toward a target in this case is really more of a mental focusing exercise — you'll note that Vader had no problem strangling Ozzel without lifting a finger. There are examples of telekinesis within the films with and without gesturing, strongly suggesting that the gestures are not a prerequisite but rather indications of concentration or even personal style.
Powers like Force lightning, however, actually do use the limb as a conduit for a physical manifestation of energy. The Force may be what creates the lightning, but it's the limb that safely directs it. Imagine, cinematically, trying to convey Force lightning without the use of gestures. Where would it come out of? How would it be aimed?
The exact science behind such power will probably never be delineated in a guidebook or publication, but George Lucas himself has stated that Anakin cannot create lightning with his mechanical limb. Now, this direction needs to be reconciled with some rare but definite precedents that seem to contradict it. For example, LucasArts' forthcoming Episode III video game features Anakin blasting away with Force lightning with his mechanical arm. But that can be excused as a convention of game mechanics and player wish fulfillment. The bottom line: it makes for a cooler game.
Tricker to reconcile is this scene in Splinter of the Mind's Eye, the first spin-off novel written well before any sort of Force "rules" were ever establised: As Vader drifted slowly back to the floor, he grabbed his right wrist with his left hand, made a fist, and seemed to convulse like a man retching. A ball of pure white energy the size of his fist materialized in front of Vader's hands and moved down toward the wide-eyed Luke.
Though the comics adaptation of Splinter, published in the mid-1990s, depicts this action with something that looks an awful lot like Force lightning, the energy that Vader hurls isn't exactly that. It is pure kinetic energy — the ball that strikes Luke is called a kinetite, or "restrained energy globe." It's possible it's not even a result of a Force power at all but rather a device that Vader built into his suit.
- Star Wars Insider Magazine Issue 80, Ask The Master