Is it possible to control the Internet by microcontrollers?

Imagine the internet as a library and each website is a book. Rephrasing you question, can we keep books at home and not only at the library? YES.

Microcontrollers are the same as a server or PC however it has different components (RAM & hardrive & ...). Due to that, they can be programmed to deliver a website as a response to HTTP request.

However, since the microcontrollers are slower and with less capacities, they will only deliver smaller websites but you can speed them up with additional external memory if that is your Goal.

So the answer is yes the internet can exist on a network of microcontrollers. Proof: i did a website on a PIC16f