In Iron Man Obadiah steals Tony's arc reactor which leaves him with the Mark 1 chestpiece (which he puts in his body so he can survive).
If he only has one arc reactor, and he's using it in his body, how does he manage to use his Mark III suit?
The arc reactor in his body is the one that is powering the suit. He doesn’t need an extra one when that is already doing the job. The one in his body is only essentially an electromagnet and doesn’t use up much power so there is a lot left available to use to power his suits if needed. This is made somewhat explicit it Tony’s initial escape.
Yinsen: That could run your heart for fifty lifetimes.
Tony: Or something very big for fifteen minutes.
He later has arc reactors for his suits independent of the one in his chest because it makes more sense. It means the suit isn’t drawing from the vital power of the one in his body and it can use more power itself. However, that doesn’t mean the one in his chest can’t power his suits just that he chooses for it not to.