The WebDesigner component contains the Save and Save As buttons on the toolbar. However, they are hidden by default.

To enable the Save and SaveAs buttons of Web Designer's toolbar, you should use the code as shown in the example below:

GrapeCity.ActiveReports.Designer.create('#ar-web-designer', {
            appBar: {
                saveButton: { visible: true },
                saveAsButton: { visible: true }
            }       
        });

The process of saving the report is performed by the UseReportDesigner() middleware:

app.UseReportDesigner(config => config.UseFileStore(ResourcesRootDirectory, false)); 

The report is saved in the Resources folder of the project.

The saving process is performed on the server side and you can customize it with the custom store. See the WebDesigner_CustomStore sample for details.