I just returned a digital camera that I had bought on QVC for this exact reason. Unless the subject was sitting completely still, I'd miss the shot because it took the camera SO long to snap the picture after I pressed the shutter button. I was really ticked off because they made it look so great on the on-air presentation, but after I thought about it, all the shots that they showed were still shots. Conveniently, though, they never mentioned the problem with the shutter delay.
I am not an expert in the field of photography so I cannot speak with expert knowledge, but from what I have been able to find out from researching on the internet, digital cameras do have this problem. You have to do alot of research before buying to find out which cameras have the best shutter speeds. I did read that manufacturers are working on improving shutter speeds since this is a common complaint, so maybe we'll be seeing an improvement soon.
A digital SLR would be fantastic, but unfortunately I can't afford to spend $1000+ on a camera right now. I've just decided that I'll stick to my Canon Rebel 2000 for a while longer until new digital models come out that downplay this problem and prices come down.