I probably need outgoing proxy or something similar in Azure to have the flexibility of vmss to scale-in my stateless app by changing the number of instances (autoscaling) and use a single public IP address with no limitations on number of IP connections (that is the case for basic and standard load balancer). How can I attache a nat device to a vmss in azure? Probably that is the wrong question. What is the right questions.
You could use an Azure load balancer as a proxy for outbound internet connectivity by using source network address translation (SNAT). See how to deploy VMSS with existing LB and outbound rules and guidelines should be considered. If the backend VMSS don't have instance-level public IP (ILPIP) addresses, they establish outbound connectivity via the frontend IP of the public load balancer.
If you are facing some issues, see how to throughout the outbound connection.