I don't understand the requirements but I'll just assume they are there. I recently did some work for a doctor's office who had some frustrating legacy requirements. Thankfully the software (server and client) can be installed/configured by anyone not just the software vendor, the full installation package was available, and runs fine in XP Mode under Windows 7 Professional 64-bit.
So assuming you CANNOT reinstall the application(s), is a major potential limitation. You can try swapping the hard drive to different motherboard and chipset but there is NO method to guarantee it works unless it's an exact motherboard replacement (or motherboard that uses a common/shared BIOS build).
If you want to try it, keeping in mind that you may be stuck paying for the hardware you can't use if it doesn't work, then this is what I would try personally if I were in this situation:
ASRock 775I65G R3.0 LGA775 Intel 865G + ICH5
Uses the same 865G chipset as the board in the Dell but has LGA775 socket that supports Core 2 Duo generation processors. Brand new, but only supports 2GB RAM MAX. It's worth noting any boards dating back to 2005 ~ 2007 production could be affected by the "bad cap" electrolytic capacitor debacle that plagued motherboards from those years. This board is very recent manufacture probably in 2011 or 2012.
If you should give it a try, I would make an image of the present hard disk so that you can restore it if needed. And test restoring the image first to a different/spare hard drive to make sure there is nothing wrong with the disk image, restore utility, anything with that process.
Basically you want to remove/uninstall several devices or their drivers on the present system prior to swapping boards. e.g. uninstall the audio, LAN, and 1394 drivers and then disable it in BIOS (if integrated), or remove from the PC if add-in. Uninstall the graphics driver and change to Standard VGA adapter. Uninstall any Intel storage driver for the ICH5 integrated IDE and SATA controllers, use the in-built driver provided by Windows XP. Uninstall printer drivers/software and disconnect. And then pray that Windows negotiates the hardware changes gracefully, including upgrades the HAL type from Uniprocessor to Multiprocesser (if you use a dual core CPU).
There is also the in-place upgrade (reinstallation or repair install) of Windows XP that will attempt to preserve the functionality of any applications. I say "attempt" because that's not always successful, either, and has some of it's own peculiarities like having to use installation bits that are patched to the same service pack level presently installed (e.g. if SP3 is installed, the installation bits need to include SP3) and having to uninstall Internet Explorer versions later than IE6 prior to reinstallation because they may not work after.
Are you sure they can't just purchase a new PC with good specs and Windows 7 Professional 64-bit, then have the software vendor install the application in Windows XP mode virtual machine? How much more could that cost (in time, potential headache, and actual monetary cost) than what you want to do?