Domain controler on Azure VM

Yes, it's possible, but if the VPN connection goes down you will lose all domain functionalities in your office, including all authentication and even DNS resolution (which would also make your Internet unusable).

I'd strongly advise against that.

Is it possible? Yes.

Is it advisable? Not really.

You should maintain at least one Domain Controller on premises in order to authenticate domain joined clients, and to provide services like DHCP and DNS in the event that the connection to Azure is unavailable.

You might also consider how you'll provide access to file and print services if the Azure connection is unavailable.