There seem to be lots of headphones around that use a Lighting connector, but is there a cable or small adapter to allow USB-C headphones to connect to an iPhone's Lightning port?
As I understand it a regular USB-C to Lightning cable won't work (although I don't have one to try it). I am aware of Apple's large USB-A adapters but they are unwieldy compared to a cable.
Considering that many USB-C headphones are designed to wire directly to an analog sound circuit in the phone they're designed for, I'm not sure how well that would work. The Lightning to headphone adapter is actually a tiny USB ADC/DAC.
Also many USB-C headphones have a DAC though. I would say probably most, unless you get a specific phone OEM branded one whose phones don't have digital output from the USB-C port. This was a problem some time ago, but I'm not sure how much of an issue is now. FWIW all Google Pixel phones don't support analog through USB, they all do digital.
Regarding the OP, the easiest way to solve it would be to just get a Lightning headset and then get a female lightning to male USB-C adapter. I've done that with the Libratone Q-adapt lightning (https://www.amazon.com/Libratone-Adapt-Lightning-Cancelling-Headphones/dp/B01N6UAWTY) and the Anker adapter (https://www.amazon.com/Anker-USB-C-Lightning-Audio-Adapter/dp/B07R6MKJZH/).
I think the reason there appears to be no female USB-C to male Lightning adapters is the same reason there are no "legal" female USB-C to male USB-A or B adapters. Since USB-A and B (and Lightning) are directional, it could be dangerous to be able to build a straight cable (not hub) that's A-to-A, B-to-B or L-to-L. You can build L-to-L with the Camera adapter but I think that behaves more like a one-port hub than a straight adapter, similar to USB-OTG which is another specific exception.
Don't know how good it is. Some comments say it works, some - it doesn't.