In fact, every machine I have used to connect to my DM all have onboard / integrated FW interfaces.
Of course, what works best for you, is what you should chose. But let me bring up some points why this might not be the best of ideas:
1) Not all mainboard manufacturers chose the best of chipsets for their firewire sets. And even if so, I've seen some pretty clunky stuff out there across all brands (ASUS, Gigabyte, Fujitsu, MSI - you name it). That doesn't mean all of them are bad, but chances are you have a chipset that isn't working properly or the overall embedding is quiet unfortunate.
2) Mainboards with onboard firewire, seem to be about to fold. There are still some out there, but I am not sure if they are on boards that support the latest iteration of CPUs (like i8, i9). I highly doubt it, but to be honest, I didn't check this.
3) A PCIe or PCI card has the big advantage of being replaceable. The drivers are specifically for your card, and should your card be no longer supported or even break, you can get a new one quiet easyly. If your mainboard breaks, is no longer supported or your onboard firewire interface doesn't do it's job properly, you're basically lost.
As I said before: there is nothing wrong with going "onboard", as long as it works for you - but IMHO there are some good reasons why you would maybe chose otherwise.