Is it possible to change the app settings for a website from the app itself?
This is not meant to be an everyday operation, but a self-service reconfiguration option. A non-developer can change a specific setting, which should cause a restart, just like I can do manually on the website configuration page (app setting section)
In addition to Diego answer, to use the Azure Management Librairies within a WebApp (WebSites and/or WebJobs), you need to configure SSL which is a little bit tricky:
Using Azure Management Libraries from Azure Web Jobs
Have you read into the Service Management REST API? The documentation mentions that it allows you to perform most the actions that are available via the Management Portal programmatically.
You can also use the Azure Fluent Api.
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It wasn't that hard once I found the right lib to do it, Microsoft Azure Web Sites Management Library.