Thanks for the advise. A quick update: Phoenix42 and I chatted on the phone and he was a huge help. The server is running VMWare ESXi 5.5, and I can get to the console. The problem: the client's former IT company didn't give them THAT login. I can only guess that they decided to take a sneaky way of making themselves indispensable. Gross. Now they're claiming they don't know what that password is, and the tier 1 tech we talked to didn't understand that an external drive had to be added to the VM in question. I couldn't even explain that fact to him in a way he understood. I don't know this stuff and even I could grasp that concept as canuckInOR informed me.

Anyway, to answer your other questions: I can access two of their VMs via RDP. The admin one that runs Active Directory, and the file/print server.

I've finally resorted to transferring the files to another computer on the network like you and Phoenix42 suggested to me. It's really slow going, but it's working now. I was having trouble getting more than about 60KBps earlier, but now I'm up to around 9-11MBps on average. Not great for 400GB of data, but I'll see what I can do...