How can I customize vehicles I can't customize?

If you're playing on the PC, you can install the Extended Customization mod which will allow you to customize all vehicles (including planes and helicopters) - even those not allowed to be customized by the vanilla game. Note that this mod is already part of the Things To Do In Steelport mod compilation (which I recommend you install, instead of the standalone version). That mod compilation also contains other game fixes and improvements.

Screenshots of vehicles customized with Extended Customization:

There is a glitch you can abuse that will allow you to customise vehicles that are tagged as non-customisable.

First, leave the vehicle you wish to customise in a garage. Then, take any customisable vehicle to a Rim Jobs. Once you get into the customisation menu, press Garage and highlight the vehicle you want to customise. Without actually selecting the vehicle, exit the menu, and answer in the affirmative when it asks you for confirmation. You will now be in the customisation menu for the vehicle you highlighted.

Edit: According to this video, this glitch will not work with all vehicles driven into the Rim Jobs; only some will work for each, with the exception of the Gat Mobile, which will allow you to customise any vehicle.