This should work fine, assuming your new Dell has a PCI slot on its motherboard. Pretty much all motherboards still do due to the wide range of PCI hardware still on the market.
As for Windows 7 support, it is quite likely that Windows will have its own driver for it. Even from my experience of reinstalling Windows Vista on a wide range of hardware and brands, I don’t recall ever having to reinstall a 56k modem driver despite most of them still with a PCI 56k modem or internal (for a laptop), so I would imagine the same for Windows 7.
It’s mainly printers and especially flatbed scanners that can be tricky finding Windows 7 drivers for.