Depending upon how Old the system is, You can program with a laptop, Always from a phone.
You know, something ' always ' changes when a customer Moves from one location to another(even if they say ' no changes ') So, If they need to change anything You will need the passwords. There are 2 levels of passwords , Installer and Admin.
Again, all the programming is held on the Harddrive(or with the newer systems on a Flash Drive) All the recordings, all the voicemails, Everything.
So, If you were able to find something ' cheep ' It would have to have Their programming in it.
' zero ' downtime? maybe you can set up a couple of slt phones, forward their lines? Move the system, and unforward? Just a thought.