Although it seems impossible with current technology to connect organ's capillaries to the recipient's capillaries and its motor neuron to the CNS, transplanted organ works for years. But why?
Capillaries only transport blood for short distances, and within an organ. The blood supply to/from a major organ is generally carried over a few major blood vessels (usually one artery, one vein). Splicing such larger blood vessels are what surgeons earn their pay to do.
In terms of nerves, most transplanted organs function fairly well without being connected to the CNS.