iPhone doesn't charge in car but does at plug

Question - have either or both car charging supplies worked in the past so this is a new problem, or always been a failure?

The first thing I would try is a different Lightning cable, genuine Apple (examples at Power & Cables - iPhone Accessories - Apple) not some kind of far eastern counterfeit. They might be slightly more expensive than the clones, but they come with a guarantee.

It might sound unlikely but the constant handling and flexing at the plugs can sometimes cause intermittent problems, and even the USB connector can go flaky. Give the USB plug a good workout, connect and disconnect about 6-10 times to give the sliding contacts a good scrub. Look into the Lightning port on your phone and clear any lint which might have got pushed in from your pocket (blow or a fine wood or plastic toothpick, not metal).

Apart from that it may be the plugin adaptor is defective or not recognized by iPhone, so try it on a different device to be charged if possible. (sometimes the plug tip on these unscrews to reveal a fuse, check it). Worst case you need to buy a good quality car adaptor, so consider the offerings while you are looking to order a new cable.