Please visit https://linear-discord-serverless-pink.vercel.app/ which documents the setup. In short, we form a URL that contains the Discord webhook ID and Token, and use that as our linear URL. That way we can use the body with ID and Token in a stateless environment.
I've made a small video guide to visually demonstrate setup. You can watch it on YouTube.
This project is heavily inspired by @ezolla's amazing linear-app-discord. If you're looking for more control over the data sent, I recommend checking that out.