In short, no.
Ignore Chugalug_, he doesn't understand what he's even saying.
Anyway, a soundcard is the most important piece of audio equipment in a computer.
An Audigy is a very nice card, I use a 12 year old Audigy 4, which was $170 new. Break out box and remote control still work.
My sound output is many many times better than any onboard audio, and you don't need amazing speakers to hear it.
Those headphones have their own built-in sound card. So they will not be able to directly use the analog output of a dedicated sound card.
Search for a 3.5mm headset to use with a dedicated sound card. They run (heaps) cheaper and typically contain speakers of much better quality that the expensive usb headsets, where all the money is...