Hello Luke,
"I don't know the vcenter nor the ESX root passwords."
This is the major hurdle here - if you had these it would just be a case of Host Client/SSH and powering stuff up - does anyone else in your organisation have these and/or might they be stored somewhere you can access?
If this is not possible then the only feasible way to change the root password of an ESXi host is to re-install ESXi - if you can re-install a host and configure access to the required datastores then you can register and power on the VMs required to access everything else.
"Everything here was extremely well planned for disaster events"
I wouldn't really be saying that at this point as you appear to be locked out of everything
Bob