We have a VPN set up that is a little troublesome. It doesnt assign a gateway IP address (a separate issue) but most annoyingly, I cant ping machines by their computer name, I have to use their IP address. Why?
A VPN setup that doesn't assign a default gateway address is OK - this just means that traffic not directed towards hosts that live on the VPN will instead traverse your normal Internet connection. Some work VPNs are set up this way - browser traffic, for example, won't go through the VPN if the default route isn't a VPN host. Some work VPNs prefer to set it up where all your Internet traffic is routed through the VPN. It can be done either way.
Many LANs have a local DNS or WINS server to resolve names for local machines. For example, at my work, I can type
ping ws1000 to ping the computer named ws1000. This is because on the LAN, my DNS is pointed to our LAN's DNS server (which is also our AD server).
On the Internet at large this won't work because only the work LAN's DNS server is setup to return a specific IP for the name
ws1000. Once I hop on my work's VPN, which "takes over" the Internet connection as in the second case above,
ping ws1000 works fine.
So you need to first get your DNS servers visible on the VPN. Then, if your LAN's DNS servers are accessible via the VPN, you need to configure your VPN software to tell Windows to set the VPN connection's DNS servers to those. Now, this does mean all DNS queries will be going over the VPN, but it's what you need to do. Of course your LAN's DNS servers need to forward non-local queries to an external DNS server such as OpenDNS, Google, or those of your LAN's ISP.