CA, like many states, requires you to have a license to moonlight - it would be hard to function in most moonlighting gigs without one. The difference, as described above, is that CA requires 36 months post-grad training to get your license. It used to be 12 months, so I got mine at the end of intern year.
The other complexity to this question is what is meant by "moonlighting." E.g. we have scanner babysitting shifts at my institution that all rad res years do. It's internal, pays well, and you can use most of the time to study, do research, watch Netflix, etc. In order to do radiology moonlighting where you're actually reading and signing scans, you need to have passed boards (e.g. R4 year/fellows) outside of rare institution-specific things. In terms of general medical moonlighting, e.g., at an urgent care clinic, you just need a license.