Is it possible to update the shipping options presented in a Wix store using the API?
My goal is to get different flat rate options to be available depending on the day of the week, and figured maybe I could build a small script that runs every night that updates the shipping options?
You mean change shipping options in your dashboard? There is no way to do it because Wix elements and components (like Wix Stores and shipping because it is dependent on Stores) are proprietary, thus the only control you have over them is what they give you.
You could create your own alternative and then you would have the ability to customize more features.
If you want to integrate more shipping options with API - we have a step by step guide on how to work with APIs :
By following this article you can call a third-party service directly from your client-side code.