It seems there could be more than one term, to describe different flavors of this urge:
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The urge to read may stem from inherent curiosity, so "bookworm" may fit. It is not precise enough because the feeling may extend not just to books, but also newspapers, journals, educational material, Q&A on Stack Exchange, etc.
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A kind of "mania", for example a person that can't ignore advertisements, posters, or any other form of random text that catches the eye.
Best Answer
I am not trying to be funny, but perhaps simply a reader since one definition at cambridge.org includes this aspect (emphasis is mine below)