I apologize for answering your question in this way –and I tried to find ways to avoid it, but it can’t be helped: It depends on what you mean by “exist”. It seems clear that the Internet doesn’t exist in the same way that a table or a person exists –at least until you try to figure out what the difference is.
The Internet is not a single, discrete physical object like a table –it is the collective name we assign both to a communications network between millions of computers, and to the information that flows over that network. Yet one might also argue that a table is merely a collective name for a diverse collection of millions of atoms arranged in a particular configuration.
The waters get even more muddied when one considers the tradition in philosophy, primarily associated with Plato, that says that everything we perceive as ordinary reality is only a pale reflection of a truer and deeper level of existence. From that point of view, it would be hard to say whether the Internet, as a largely conceptual entity, might not have a better claim to existence than the table.