Not my field, but the most plausible candidate I've heard of is Paul Otlet in the early 20th century:

http://www.nytimes.com/2008/06/17/science/17mund.html?pagewanted=all&_r=0

http://www.brainpickings.org/2014/06/09/paul-otlet-alex-wright/

I disagree that his idea of "electric telescopes" maps clearly onto our idea of "computers," as the second article claims, but Otlet definitely had the idea of a global information network (as opposed to a global communication network).

I'd say 'The Last Question' by Isaac Asimov in 1956 was a pretty early example which suggested long-distance communication.

http://www.multivax.com/last_question.html