Docker Desktop is not for servers but for development it is not even officially supported on Windows Server. You can still automate starting Docker Desktop by allowing it to start when you log in (Settings » General » “Start Docker Desktop when you log in”).
Since Docker Desktop is a desktop application, I don’t know how it can be started without anyone logging in.
You can check the Windows Services, look for Docker and find what those services start. Mybe you can manage to start the vitual machine without the Desktop, I wouldn’t be surprised if it were not possible.
If running Docker as a server app is important to you, you could create a virtual machine, install Docker CE in it and configure the VM to start with Windows. I am not sure how that can be achieved, except never shutting the virtual machines down when you stop the host os so the HyperV VMs can automatically be restored from saved state.
Note that I usually work with Linux so most of what I say I just heard about from soneone.
What kind of containers do you want to run? Windows containers or Linux containers?