About six months ago, I was contacted by a software developer regarding "real time" inventory management, integrated procurement and order tracking. Some of these high-end solutions are fantastic but be advised that they don't come without a price tag.
I can see where a larger store might want to implement such a system but for the vast majority of smaller sites, it's probably not worth it. Given that you're a little more familiar with your clients' requirements, you'd be the best judge.
In my experience, even some of the larger electronics sites seem to maintain separate stock and, while a particular store or outlet may have run out of stock on an item, it's still available through the website, at least that's how it appears in the front end.
Here's a pretty good link to some information that might be of benefit to you and help steer you in the right direction.
The Internet in E-commerce: Inventory planning and order management
In terms of what's really required, it's all dependent on the amount of business that the store and the website do. IMHO, a few random web orders wouldn't provide enough ROI to make implementing a unified database worthwhile. If we're talking about one store and one website, it's probably not going to be worth it.
If you're doing hundreds of thousands of dollars is business a year, implementing an order management/fulfillment application might be the way to go.