First you need an existing VNet with P2S configured as per my post at http://www.techdiction.com/2016/01/12/creating-a-point-to-site-vpn-connection-on-an-azure-resource-manager-virtual-network/
Then use the below PowerShell to connect the AppService to the VNet using P2S VPN:
$subscription_id = "<Subscription_ID>"
$NetworkName = "<Network_Name>"
$location = "<Region>"
$netrgname = "<Resource_Group_VNet_is_in>"
$AppServiceName = "<AppService_Name>"
$props = @{
"vnetResourceId" = "/subscriptions/$subscription_id/resourcegroups/$netrgname/providers/Microsoft.ClassicNetwork/virtualNetworks/$NetworkName";
"certThumbprint"= "<Client_cert_thumbprint>";
"certBlob"= "<Base64_Cert_Data>";
"routes" = $null;
}
New-AzureRMResource -ResourceName "$AppServiceName/$AppServiceName-to-$NetworkName" -Location $location -ResourceGroupName MarcusWebsites -ResourceType Microsoft.Web/sites/virtualNetworkConnections -PropertyObject $props -ApiVersion "2015-08-01" -force
You can configure custom routes if you require by modifying the routes property. Let me know how you get on and if it resolves the situation please mark this post as the answer.
Marcus