Does anyone know if Azure API Apps support WebJobs? If so is it possible to deploy them to Azure from Visual Studio 2013?
I've managed to get an API app pushing to an Azure Queue and a WebJob triggering on that event with the code running locally on my development machine (i.e. accessing the queue in Azure rather than on the azure development emulator), but can't publish the WebJob to Azure and also there's no WebJobs blade/option from settings in the API App entry in the Azure Portal (i.e. the preview portal)
You can add anything that would work on a WebApp to an API App as well. An API App is being hosted in a WebApp.
To find all the similar settings, including WebJobs, you have to go to the API App blade in the portal and then from the Settings on the top (Essential Settings) find the one that says "API App host". Click that one and this will take you to the WebApp blade that is hosting the API App. There, if you click "Settings" from the toolbar, you should get all the WebApp settings, including WebJobs, custom domains, SSL etc.
Once we release the ARM template customization for the API Apps, you could use it to deploy the WebJobs and all the other customizations as part of the API App deployment.