How can I take 360° panoramic pictures in an aircraft or helicopter?
This is as much a photography question as an aviation one. The type of aircraft you might want depends on the type of picture you want.

If you want a picture from low altitude, showing the area nearby, a helicopter or a quadcopter (depending on local regulations) would be ideal. This would let you hover in one spot, and rotate to cover the full panorama. The movement of the aircraft could easily be compensated for in processing the photos. A quadcopter holding a position by GPS should be very stable for long enough to take the photos.

If your picture is at a higher altitude, focusing more on things further away, a small plane could work. Many general aviation aircraft can fly fairly slow. With only more distant objects as the subject, a plane in a turn at under 100 knots should be slow enough that the perspective change would be negligible.

If you could get something like a simplified version of Google's street view setup, and send it up on a quadcopter, that would probably give the absolute best result.

See also this more general question:

What is a good plane for aerial photography?