You can use Powershell to trigger Manual Azure Webapp Webjob.
The user name and password comes from their Publishing profile. (In the Azure Portal -> browse to Your App Service > Click on "Get Publish Profile")
The $username in the script should look like SomeUserName, and not SomeSite\SomeUserName
$username = "`$username"
$password = "password"
$base64AuthInfo = [Convert]::ToBase64String([Text.Encoding]::ASCII.GetBytes(("{0}:{1}" -f $username, $password)))
$userAgent = "powershell/1.0"
$apiUrl = "https://<sitename>.scm.azurewebsites.net/api/triggeredwebjobs/<YourWebjobName>/run"
Invoke-WebRequest -Uri $apiUrl -Headers @{Authorization=("Basic {0}" -f $base64AuthInfo)} -UserAgent $userAgent -Method POST -Debug