The campaign - Sprint for Shakespeare - aims to raise £20,000 to put 1,000 pages of the playwright's work online. This equates to around £20 per page.
The appeal has already won the backing of stars of stage and screen, including Stephen Fry and Vanessa Redgrave.
Once the project is complete, members of the public will be able to access the website and the plays free of charge, the Bodleian said.
There will also be articles and blogs from academics, specialists and theatre professionals.
The Bodleian said that, while copies of this book were not rare, its First Folio was a rarity because it had not been rebound or restored in nearly four centuries.
Sir Peter Hall, founder of the Royal Shakespeare Company and theatre and film director said: "The digitisation of the Bodleian copy of Shakespeare's First Folio is a project of huge importance.
"It will provide an unrivalled opportunity for textual study not only for actors, directors and other theatre practitioners and their academic colleagues, but also for audiences whose love of the plays has remained undiminished over the centuries."
'Madness and hilarity' Stephen Fry said he backed the project "whole-heartedly".
"First Folio as a phrase sounds so distant from our everyday lives, but this priceless and extraordinary collection of plays turned the world upside down (or should that be the right way up?) every bit as much as Newton was to do nearly 60 or so years later.
"The works of Shakespeare, now as much as ever, tell us what it is to be alive.
"The ambiguity, doubt, puzzlement, pain, madness and hilarity of existence had never been expressed so well and to this day never has.
"To bring the First Folio, the great authoritative publication, to everyone in the world via digitisation is as noble and magnificent a project as can be imagined."