Attaching electronics to a spinning surface without a battery
You can use contacts (e.g. carbon brushes, or copper) to press lightly against traces on the spinning disk and transfer power. Probably the simplest way. 
Or you could possibly use inductive coupling, e.g. something like a toroidal power coil on the base transfers power to a toroidal coil on the disk, although this would require more thought.

A good source of ideas would be the numerous POV displays everyone seems to be building (one row of spinning LEDs make a picture) I'm sure I have seen some inventive ways of supplying power (aside from battery) in a few of those. Google and You Tube will know more.