I heard that some people had to call NVC to get their case number. In most cases they already case numbers assigned but for some reason they don't send email or call. So maybe try to call their number especially early in the morning. However be ready to wait for more than 30 minutes with all your information ready

we have called them almost every day for a month. Every time they say “we have not received this yet” 🤷🏻‍♂️

You will receive your case number and invoice ID when your case is sent over to the embassy. But, you can continue calling them to receive that information.