Update: Got it fixed.
Needed to use API key from specific user, not the general API key. To configure, click your user image in the top left, click the gear for “Preferences”, click “Admin”, and then set up an API key there.
Category IDs (for specifying where the post should go) I found here:
http://mydiscourseinstallation.com/site.json
Zapier setup looks like this:
All other fields I could leave as they were.
Works like a charm now! Thanks guys.