No, the internet you have is conveyed via radiation, of which the speed is equal to the speed of light, or through wires by moving electrons, whose drift velocity is less than the speed of light. (This is obvious since electrons have mass. According to special relativity mass tends to infinity at light speed. Hence no amount of force can cause acceleration to overcome $v=c$.)
Unless you figure out how to use tachyons for communication. But that is also not possible I suppose, as tachyons don't interact with ordinary matter. And as a direct result, no observation of them is possible, as far as i know.