In my experience, once you've won an auction, the buyer will contact you no more than a day later and inform you of your payment options etc. If they don't then send them a Message via GunBroker in the auction summary. I haven't had any problems and I've done about 20 or so buys and a couple of sales on there.
Pretty much this... I'll add that it's a pretty safe venture, considering there is a feedback system in place. The only people I tend not to do business with are the ones with no feedback (NR)
- Clint