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.

The lowest tech, probably easiest method - Use brushes, just like for a motor. On the rotating part, you'd have the equivalent of slip rings, although the orientation would be flat rather than axial. The power thus obtained on the rotating part might have a bit of high frequency noise in it from the sliding contacts, so you'd want to be sure to get some filter caps on involved, probably near to where the power comes onto the board.