The drone API reference does not provide info about an orgsecret endpoint. However, I suspect there must be one as the drone CLI might most likely use it via the drone orgsecret subcommand.
Having an API endpoint would simplify automation efforts via ansible or similar.
NB: I am aware that the endpoint only works for drone admins.
@pat-s yes you can use drone orgsecret add subcommand to create new organization secret and yes it requires system administrator privileges. Here is the in detailed documentation for using different subcommands on orgsecret
You will find troubleshooting guide for the same over here