Cover carport framing
You could paint it white without covering the walls, and solve both the dark and potential permitting issues with the other approach. Paint is generally free of permitting needs (so long as you source it locally so it's VOC legal for California/LA air pollution issues, I suppose from afar.)

Permit/code issues would better be addressed by a local - as an interior low-cost project that doesn't change any doorways/windows, or add a structure to the property, it would probably be fine without a permit here. Cities tend to be a bit more irritating and/or want more pounds of flesh, so I'd guess you might well be in need of a permit given that you are in a big one. When in doubt, ask, as the consequences for doing without are usually a lot harsher than the hoops to jump through to get one. And you might actually find that you don't need one - but that's a local jurisdiction having authority issue, not something the internet can reliably answer for you.

Material-wise will depend on your local bug population and why the INSIDE of your carport (with walls, so presumably not a "roof-only, classic" carport, thus creeping closer to a garage) is subject to moisture damage. You might want to use something like tile backer board (Durock, Hardibacker, etc.) for the bottom 3 feet (it comes in 3x5 sheets) as being pretty much immune to bug and water damage. If it gets wet higher than that, either stick with the same product or choose an exterior-rated plywood. Tile backer is also nicely fire-resistant. If it's dry inside, drywall should be fine, as it is in many garages. A picture might help.