@COSPNWGuy @ISP_graynic @ZachBodenner @Blevins_Mark - That is correct. The ArcGIS Enterprise connector should be available in the government tenant (apologies for the slow delay in getting it deployed). We are currently in a deployment process to have triggers set up in the ArcGIS Enterprise (purple) connector that mimics the same behavior as the ArcGIS Online connector (orange). Meaning, you won't need to go through steps listed in the link below in 11.2 and above.
Once those have been deployed, you can configure triggers as you would in the ArcGIS Online connector as long as your Enterprise instance is on 11.2 or higher. If you are on 11.1, then you must set up triggers and webhooks using the steps in this guide: Enabling webhooks and Fetch Updates for ArcGIS Ent... - Esri Community
As for the documentation link that @ISP_graynic posted; it is slightly outdated. All actions that are available in the ArcGIS Online connector are available in the ArcGIS Connector. We are working to update that documentation page in our next release.
Let me know if this helps!