We usually use 'on' for the internet I'm going online I'm on the internet You can buy it on the online store. However, saying 'you can buy it from the online store" is fine.

The only difference I can think of, and I'm not sure it really is a difference, it that I might say "I'll buy it FROM the official website" if I am going to buy it FROM a particular company's website. So I might buy a Rolex watch FROM the Rolex website to make sure I don't get a fake. ON an official website might be something like Amazon. It's an 'official' website but handles goods from many different companies. So I might buy a Rolex watch ON Amazon, FROM a particular supplier. On the internet, on Amazon, on Ebay, from a particular company or supplier or store. But there are a lot of situations where they might be interchangeable.