On the vCSA VMWARE_DATA_DIR maps to /storage.
Here are the permissions for the storage folder:
drwxr-xr-x 17 root root 4096 Nov 9 09:25 storage
If you go to /storage and check it content:
drwxr-xr-x 3 vsphere-client root 4096 Nov 9 09:26 vsphere-client
The Web Client server is running with user "vsphere-client" and it has write permissions to /storage/vsphere-client folder.
sc-rdops-vm18-dhcp-48-250:/storage # ps -u vsphere-client
PID TTY TIME CMD
17669 ? 00:00:00 wrapper
17678 ? 00:00:10 vsphere-client
Hope that now it makes more sense why you cannot write directly to VMWARE_DATA_DIR
