Hi Graci,
Currently LabVIEW for Linux requires an X server. You can accomplish this by running a dummy X server on your embedded Linux PC. Or, you can install only the X libraries and then set the display to another networked X server.
You might be aware that NI has several Real Time products that might suit your needs well. You can use PXI for the equivalent of a PC, or fieldpoint if you need a smaller/embedded solution, or RT plug-in boards which are basically PCs on a PCI card. All of these options would allow you to run LabVIEW executables "headless."
Your project sounds interesting, let us know more about it and it's requirements!
David Duffey