An online shopping site is a website which can be accessed through a computer which can be either a desktop or laptop. The website can also be accessed through a smartphone or tablet and both are mobile devices. In computers, because of higher data processing capacity and more functionality, all you need to do to access a shopping site is type the URL of the site in the browser and press 'enter'. However, on a mobile device like a smartphone or tablet, you may be able to open a shopping site but won't be able to make transactions, unless you download the app of that shopping site.
The reason for this is the limited data processing and computing capability of a mobile device although some high-end smartphones have reduced the gap between computers substantially. The only problem is that these high-end smartphones are too expensive for the masses and they are always likely to remain confined to a limited number of people. Most people would continue to use affordable smartphones on which, they will need apps to run a shopping site because such websites require intense computing and data processing ability. Now, let's take up what you implied in your question - how does an app compliment as a website on a mobile device?
A mobile app is a special computer program that is designed to provide efficient access and functionality for a particular website on a smartphone where it will function just as efficiently as it will, on a computer. This app will allow you to do your shopping on that particular site in the same manner that you would have done on your computer. The user will have to download the app onto his smartphone and use it whenever needed. It's a little like bookmarking a website on your computer. You open it whenever you need to visit the site. However, if you are asking whether websites will become extinct and apps will take over, I must say that it's not likely, at least in the near future.