Are there any crimes that could apply to a site owner if his e-commerce site is hacked?
Correct me if I'm wrong, but even though the question is little detailed, I'd propose a few inputs on the matter.
First, it's important to differ crime and judicial punishment the same way we divide criminal court and civil court. It's important to note that usually a crime requires intent (with obvious exception of crimes against life); that is, to commit a crime you are required to be motivated, be interested in the outcome, to have the intention to do it. On the other hand, civil courts won't bother to limit themselves to your wishes, if damages were caused, damages might required to be repared.
What we can conclude is that it's more likely for the owner to be held accountable for the damages he caused in a civil court rather than a criminal one. Although the website was hacked, the information stored there is in the owner's responsibility. The more sensible the data stolen is, the more damages it might cause.
It would depend on the jurisdiction and on the degree of incompetence and negligence that led to the hack. If password are stored, that is to me gross negligence with a significant legal risk.