The best thing to do here is to contact your IT person, otherwise you're going to have to explain the structure of your network. Are the two machines that run your LabVIEW programs the only computers on that network link, or is that part of some bigger network? If it's just those two computers, then you should assign them static IP addresses since there's no server available to assign them valid addresses automatically, and use reserved addressed (for example 10.0.0.1 and 10.0.0.2) with a netmask of 255.255.255.0. If however you have two large, independent networks, and one of your machines happens to sit on both, then you'll have to work with someone who understands your specific network. You might want to try simply swapping the ethernet cables between the two cards since your operating system may be prioritizing one card over the other. It's possible you can set up a static route between one machine and the other using arp, but that's outside my area of expertise. You should also do some experimenting with ping in a command window to check basic communication between the two machines.
I do currently have a system running with two network cards, one talking to several control/acquisition devices and the other to our corporate network, without any problems, although it's not two LabVIEW programs talking to each other. The acquisition devices and the card that talks to them all use reserved internal addresses (10.0.0.x) independent of the corporate network subnet.
