Is a license required to transmit in the designated aviation band?
The FCC does not presently require a license for Aviation Radio installations in aircraft, where those are VHF and UHF (as I recall).  These are covered in Part 87 of the FCC regulations.

Travel outside the US requires a station license, and I believe HF transmissions require a license.

Similarly the operator (pilot) needs a permit when traveling outside the US, but does not need the permit within the US, to answer your question about the operator.