Unless you have the actual php source from this website, there really isn't a way to be 100% sure what shopping cart system it's using. That is the great thing about these shopping cart engines, they can be modified and tweaked. From a quick glance, it appears to be a template based on the oscommerce/zencart/CRE cart system.
If you can get your hands on the original php source, it would help. Good luck with that though!
It sounds to me that you can accomplish everything you need with a simple oscommerce or zencart solution. Even if the original source does not offer a specific feature you are looking for, I guarantee someone out there already wrote the PHP code for it. For example, If you need an ordering calender system for oscommerce, just google "oscommerce booking calender". Easy modification is what has made these systems so popular.
Good luck! Hope this helps