When you capture the Affiliate code, scraping it out of the URL using Javascript in your theme, you can save it as a cart attribute. Those get passed to the order, and are available for the lifecycle of an order... start to finish, provided a cart exists. And those are not a given until someone puts something in the cart, so you can just save the affiliate code as a cookie till the cart exists and then once it exists, persist the affiliate code as a cart attribute.
You're on your own here, as you can make up dozens of ways of playing with Shopify and this kind of processing. Experiements will lead you to the best choice for you.