I connect to a client's network via the Cisco VPN client. Once I connect, my local network access (including internet access) is disabled.
Is there a way to get around this, besides for them to change something on their end (because that will not happen)?
You could set up a virtual machine (using virtualbox or something similar), and use that to connect to the client, leaving your main connection open to the internet.
Pip is correct about the remote default gateway. This is a specific choice made by the VPN administrators. It is sometimes for connectivity reasons -- ie connecting to non-local systems across a corporate network -- and it is sometimes sold as a "security" feature, in that systems connected to the VPN either cannot contact the internet or must do so through the corporate internet presence where presumably such activity can be logged.
I am not aware of any circumstances where fiddling with the local routing table gets any lasting effects -- you would probably have to write a script to to repeat the fiddling during each connection to the VPN.
Brent's solution is perhaps the easiest. I personally have a VM for each customer network just so I can be connected to multiple VPNs (mostly SSL-based, but some Cisco clients too) at the same time without having to worry about conflicts.
This is probably disabled by the VPN policy and is called dual-homing. If this is the case, then it may not be possible to get around short of using a separate machine -- even a virtual machine will be trapped in the tunnel.
Dual-homing is usually disabled to prevent your machine from become a gateway for the transmission of virus, hacking or other nasty bits.
I have a similiar issue.
My current work-around is to use a different laptop, which uses the original internet connection with no problems.
From this I would guess that a virtual machine connecting to the client machine, and having it's network settings modified, wouldn't impact the base machine.
Try going into Advanced Mode (Ctrl-M), go to the Options menu, and uncheck Stateful Firewall (Always On). This has sometimes worked for me.
I know this is because your PC has to change its default gateway to be a valid gateway on the remote part of the VPN. If it didn't do this then you would be un-able to contact any device through the VPN that wasn't on your local subnet. You should be able to keep access to your local subnet as long as the address range in use doesn't clash with any address range on the VPN.
You might be able to fiddle with your IP settings, set up static routes on your PC so that internet addresses are routed through your local default gateway and only the address range on the remote network gets routed through the VPN.
When I have tried this on a work laptop I didn't manage it though. Hopefully someone else here will have a more solid answer for you. Or your Google Fu might be better than mine was. :)