I am part of a church that has started using YouTUbe over the pandemic to live stream service.
We intend on continuing this ad infinitum, however the tedium of having to schedule a live stream manually isn't ideal.
Is there a way to call an API to automatically schedule a livestream, and return the streams URL (to save into the database)?
The short answer to your question is: yes, there is a set of API endpoints that allows you to achieve your goals.
Go read the official YouTube Live Streaming API Overview, then follow its instructions.
The official reference would be of very much use during your program development: API Reference.
You did not mentioned the level of your acquaintance with the YouTube Data API, therefore, I'll also mention an important doc you have to absorb too: Implementing OAuth 2.0 Authorization. Depending of the type of application you'll choose to develop (desktop app, server-side/client-side JavaScript, etc.), you need to absorb further one of the docs referenced therein.