As for magical "battery-style" power, the quick answer is [u]no they don't[/u] and they don't need to be recharged or anything.

What they do have is a perpetual source of animate energy, which disappears when they sustain enough damage, effectively resulting in their "death" as a creature.

D&D constructs, although they are constructed like magic items are, and more than a few can produce magical effects, are classified as creatures once they are completed. They can function in antimagic fields and dead magic zones, even if some of their abilities won't work in those situations. 

However, their perpetual source of animate energy can be assaulted directly, as reflected by the [url=http://dndtools.eu/spells/spell-compendium--86/ray-deanimation--4062/]Ray of Deanimation[/url] spell, despite the language it uses.