You need a multimedia-compliant sound card if you want to use Vox Studio to record or play back. No sound card is needed at all if Vox Studio is used to perform format or coding conversions or filtering only.
A sound card is not a must for playing back files if you have another setup outside of Vox Studio to play files: over your voice-processing telephony card for instance. You may not be able to play back ".wav" files that way unless your telephony card driver allows it, and the quality may be no better than telephony quality. Vox Studio drives multimedia-compliant sound cards, but not telephony voice cards.
You should select a sound card that can do 16-bit recordings and can handle sample frequencies of 44 KHz (or more) down to 6 KHz, and do it well. Your sound card should also be able to record or playback mono signals at all the sampling frequencies of interest to you. Most cards perform fine at the standard 11 KHz, 22 KHz and 44 KHz sampling frequencies. However, where you can often really see a big difference, is how they perform at nonstandard frequencies like 6 KHz and 8 KHz or 48KHz and 64KHz. Select a sound card that is immune to the PC's power supply electrical noise.
By all means, select a card that comes from a reputable vendor that supplies quality drivers and driver upgrades, a Windows sound mixer, level adjuster and some limited graphical sound editor (to test the card's hardware). Most brand name products do.