Werewolves and vampires were occasionally connected in folklore before the 20th century, but their proper "meeting" took place in cinema. Universal Studios found critical success with horror movies in the 1930s and 40s, particularly Dracula (1931), Frankenstein (1931), The Mummy (1932), The Werewolf of London (1943) and The Wolf-Man (1941). These films inspired sequels like Bride of Frankenstein (1935) and Dracula's Daughter (1936), and also the first crossovers like Frankenstein Meets the Wolf-Man (1943) and Abbott and Costello Meet Frankenstein (1948). Dracula and the Wolf-Man would first appear together in House of Dracula (1945).
These are still, for the most part, individual monsters interacting. But the Universal Films depictions of their monsters would go on to greatly inspire and inform the basic concepts of vampires, werewolves, Frankenstein's monsters, mummies, amphibious humanoid creatures, and invisible men - and in later media, they would start to interact and conflict with one another. One of the earliest vampire vs. werewolf conflicts on this order was the comic book crossover Werewolf By Night #15 / Tomb of Dracula #18 (Marvel, 1972), which tied the werewolf's origin together with Dracula, and set them as enemies.
However, this still wasn't a concerted idea of vampires as the natural enemies of werewolves. For that, you have to turn to 1991, with the release of the 1st edition of Vampire: the Masquerade, followed in 1992 by Werewolf: the Apocalypse. These were roleplaying games which took some of the classic Universal Monster riffs (vampires, werewolves, and mummies) and reimagined them in a "Gothic Punk" contemporary setting, dubbed the World of Darkness. Heavily influenced by films like The Lost Boys and Near Dark (1987) as well as the novels of Anne Rice, the World of Darkness is essentially where the contemporary tropes of an organization of vampires ruling from the shadows originates, the back-to-nature elements of werewolves were emphasized, and the vampire-vs-werewolf conflict was codified. This setting was essentially the direct inspiration for later vampire vs. werewolf conflict in media like the Underworld film series and the Twilight books and films - but that's running directly into the 20-year rule.