Hi Kate,
The exact manner of connection really depends on what system you are trying to comment to and your overall workflows. You can use Manager to update attributes on the report which can trigger an email to be sent, or which can trigger the details of the report to be copied into another system, or which can simply flag an item for attention in the same layer when viewed in Collector.
The Workforce team has scripts that can be used to copy features from AGOL feature layers into the schema required for workforce, and we are looking to extend these in the near future. Additionally we are working with partners such as Cityworks to set up connections between ArcGIS layers and their systems.
If your client is using another CMMS, and one that cannot read ArcGIS layers directly, I suggest looking at the ArcGIS API for Python to export the contents of the reports feature layers into a format that can be read by their CMMS.
-Allison