After a bit more digging, I found out that this site was running vCenter Server 5, NOT vCenter Server 5 Update 2.
I've update the site's vCenter Server installation, but the issue remains.
The build for vCenter Server is 913577 (5.0 Update 2)
The build for the host is 1254542 (5.0 Update 2 is 914586 according to VMware KB: Correlating vCenter Server and ESXi/ESX host build numbers to update levels).
The VM is Virtual Machine Version 8
The guest OS is Windows Server 2012 with VMware Tools version 8.6.11, build 1180212 (vSphere reports it as current)
The host has been scanned for updates and remediated. Both the VM and the host have been restarted. vCenter Server services were stopped and started as part of the update process.
Whenever I attempt an application-consistent snapshot, the resulting vss_manifests zip file that gets created has only a single file (backup.xml) with a single line. There are no files for the individual vss writers.