Alec Coppel and Samuel Taylor
Vertigo is based on the 1954 novel by Pierre Boileau and Thomas Narcejac, D'entre les morts, translated into English as The Living and the Dead. Paramount was hot to get its hand on the novel, and they purchased the rights before it was even translated into English (source). The French title means literally From Among the Dead, and—fun fact—this was the working title of Vertigo when shooting first began.
The two writers credited with Vertigo's screenplay are Alec Coppel and Samuel Taylor. The first stab at the adaptation was by playwright Maxwell Anderson, but Hitchcock disliked the result so much he told his producers to "burn it" (source).
The next two writers came into conflict over who'd be credited with Vertigo's screenplay. Coppel had worked out many of the specifics of the adaptation, but Taylor was the one who gave the film's dialogue its final form. Besides hammering out the zingers we know and love—lines like Madeleine's "Only one is a wanderer. Two together are always going somewhere"—Taylor made a crucial addition to the film: he created the character of Midge, who represents good humor, optimism and sanity in a film filled with its opposite.
In the end, both Taylor and Coppel, who'd worked on the screenplay earlier, would receive credit for their work on the adaptation.