Is there a way to trigger a job run via the API? I see things to list, cancel get information about them, but nothing that looks like a way to kick one off.

yes it's possible: https://airbyte-public-api-docs.s3.us-east-2.amazonaws.com/rapidoc-api-docs.html#post-/v1/connections/sync

you need to send the connectionId (the uuid in the URL)