I know this is meant as a provocation, but I'm going to take it seriously anyway.
The artifact that matters isn't just the prompt, it's the entire conversation.
It's rare when vibe coding to get exactly what you want first time. Usually you'll need to reply with follow-up prompts - requests for small or large changes, pasted in error messages, new ideas that come to you after you see the first iterations.
That overall conversation is, I think, an artifact that is worth preserving.
So... I link to them. Take a look at the commits for my collection of vibe-coded tools on https://tools.simonwillison.net/colophon - almost every commit includes a link to either a ChatGPT or Claude share URL or a GitHub Gist with a transcript copied from one of those or exported from my LLM command-line tool.
Sometimes I use these to test new models - I'll dig the initial prompt for one of my existing tools out and see what the latest models can do with it.
(Capturing the output is essential too, because the output of LLMs is non-deterministic - running the same prompt against the same model almost never returns exactly the same result.)