Hi Cesar and welcome to the Corona Labs community forum.
I think we need to know more about what kind of mini-games and the e-commerce platform before anyone can really offer a lot of advice.
Generally, for mobile games, they are going to be full screen on the device and they are going to be dedicated apps, not something typically run in a web browser (more on that in a bit). Then with mobile purchases, for anything that's going to be digitally delivered has to be purchased through Apple's App Store, Google Play or Amazon. If your e-commerce is for physical goods, you can integrate with your shop.
Since games are typically going to be full screen, you're likely going to present any items to sale through the games UI. You can use our network.request() API to make REST-based web server API calls. This is how you would normally have a Corona app talk to a web service like an e-commerce platform. You would download information like the item's title, description, price, image etc. and use Corona to present that information to your users.
Now it sounds like what you may be more interested in is building an HTML5 based game. This is a game that would run in the web browser and could be embedded in your e-commerce site and possibly be playable in a <div> on the site.
Games don't have typical web/application style UI elements. Sure there are concepts of buttons and lists, but drop-down menus and such are less common. Even objects like on/off switches would typically be designed to be part of the game and not be canned device style UI elements.