It sounds like the administator has not enabled splt tunneling so all traffic is going over the VPN to the internal network so you cannot access the internet sites if it is not allowed via the internal network.
"With VPN tunneling, the client’s machine effectively becomes a node on the remote (corporate) LAN and becomes invisible on the user’s local LAN; the system serves as the Domain Name Service (DNS) gateway for the client and knows nothing about the user’s local LAN."
So once the tunnel is up, the DNS servers configured on the PCS are used by the client and if they do not resolve the hostname you are attempting to contact then you will get DNS failures.