When publishing data to ArcGIS Online, you start with a geodatabase and publish that data to ArcGIS Online. Then, as the data in your geodatabase changes, those changes are propagated to ArcGIS Online through the feature service. Furthermore, changes on ArcGIS Online are handled through the same feature service interface.
If you want to have AmigoCloud access your ArcGIS Online data, it can do so by accessing the source of your ArcGIS Online data, the Geodatabase it came from. By using the geoprocessing tool you can sync the data from the Geodatabase to AmigoCloud in a similar manner to updating your ArcGIS Online data using the feature service.
If you are using ArcGIS Online to host your data, rather than maintaining it on an ArcSDE Server or a Local Geodatabase, AmigoCloud will not be able to access that data.
If you have a Web Map Service (WMS), you can then access that through ArcGIS Online and AmigoCloud as well. For AmigoCloud, just add it as a new base layer, and for ArcGIS Online you can add it as an OGC WMS.