Is the vCenter Server mentioned the vCenter Server Essentials that is part of the Essentials Kit?
Yes, the vCenter Server is included with Essentials Kit.. you will only need to download from VMware website.
Is the installation of vCenter Server destructive? Will I have to reinstall ESXi and recreate the VMs? Down time is a Big Deal for us.
No, the vCenter Server is just a piece of software for additional management capabilities of your ESXi hosts. You don't have to reinstall and recreate nothing, just install vCenter Server on a WIndows Server VM or deploy the vCenter Server Appliance (the easiest and fastest way) and add the host to vCenter Server.
For more information about vCenter Server, check the following link: VMware vCenter Server, Virtualization Management Software Features | United States