This is actually a known issue with the VDP appliance. If you create an SR with VMware support, we have a fix to resolve this issue.
If you are in a bind, look at /usr/local/avamarclient/var-proxy-? and see if there are log files you can clear up. By default we retain 14 days worth of logs, so you might be able to remove some of the older ones to free up the space.
Once the /root partition fills up, none of the services are going to start, so you will need to restart the appliance after you free up space. Long term fix for this issue will be forthcoming in a future release.
Thanks,