Despite its lack of visual feedback you can upload files directly into your Azure App Service via the Azure portal. This can be done via the Kudo Console or in the new App Service Editor (preview).
To open the Kudo Console navigate to YOUR APP SERVICE -> Development Tools -> Advanced Tools or bookmark https://yourappservice.scm.azurewebsites.net/DebugConsole. And then just drag and drop the files you want uploaded onto the relevant folder.
To open App Service Editor navigate to YOUR APP SERVICE -> Development Tools -> App Service Editor (Preview) or bookmark https://yourappservice.scm.azurewebsites.net/dev. You can upload files by dragging them onto the Explore pane on the LHS.