If you want to download your App Service web server and application log files in the Azure portal you can do this by using the Kudo console.
To open the Kudo console go to your App Service and click on the Development Tools -> Advanced Tools link as shown below…
In Kudo click Debug console -> CMD and then using the folder view at the top of the page navigate to the LogFiles folder. From here you’ll see a number of log folders including Application and http.
Turning on logging for your App Service
If there are no logs in the log folders you might not have logging turned on or might not be logging to the file system. To do this go to the App Service logs page of your App Service and turn on the logging you desire as shown below…