In ancient, medieval and renaissance sources the only way to become a werewolf is to be cursed by gods or by the use of magic, but in contemporary popular culture lycanthropy transfers by a bite. When and where did this "belief" emerge?
Sometime, I suspect, around 1935 but no scholarly work I've read has stated "this is the first reference to were-wolfery as a transmittable condition". A few more-or-less scholarly works point to Hollywood as being the source (don't ask me to name them, been a while since I read them) but they mostly emphasise 1941's The Wolf Man, ignoring an earlier film.
I've only skimmed over The Werewolf in Lore and Legend from 1933. I don't recall it mentioning transmission by bite or wound but again, it's been a while.
The same year came The Werewolf of Paris. I haven't read it myself but I gather from summaries and references to the novel that people attacked by the werewolf, a child born of the rape of a young girl by a 70 year old priest who had gone mad living in a pit for decades eating raw meat, don't transform into werewolves, mostly I think because they end up dead and eaten. The werewolf does lap blood from his lover's arm for a bit, but he stops that in case he kills her. She doesn't transform but then again she's bleeding because she cuts herself, not because she's bitten.
A couple of years after that comes the earliest depiction of transmission by wound I know of, and the first to depict the werewolf as a biped, Werewolf of London. The protagonist is bitten by a werewolf while looking for a rare plant in Tibet or somewhere around there. Later he's told he will turn into a werewolf because of that. Co-incidentally the plant he was looking for is a temporary cure for werewolfery. After that came The Wolf Man which being so popular pretty much cemented trasmission by wound in the werewolf mythos.
To clarify as per DCShannon's comment, my 1935 date is based on the movied named being the earliest work I have watched or read that mentions werewolfery as an infection. The reason I mentioned the Paris novel is that it was, to my understanding, a fairly popular piece of literature and did not mention infection. So we go from one popular work of fiction without infection to another two years later with infection and after that it seems like everything is based on infection.
Someone who cares to read The Werewolf in Lore and Legend (Montague Summers, 1933), which is not only currently in print but avaiable in ebook format may find a reference. It is rather hard going. I'd say it's fairly safe to skip over the ancient Greek and Latin bits due to stories at the time those languages were used for writing being of the this person is cursed / made a pact / cast a spell/ is a skin-walker type where there was no infection, just a lot of dead villagers or dead sheep or ripped skirts.
The Werewolf Book: The Encyclopedia of Shape-Shifting Beings (Brad Steiger, 1999) out of print. Goes off topic a lot and doesn't treat anything in depth.
The Curse of the Werewolf: Fantasy, Horror and the Beast Within (Chantal Bourgault du Coudray, 2006)
People may also like to look into the cajun legend of rougarou (cajun pronunciation of loup garou). In some versions this thing transfers the werewolf curse after 101 days but then it stops being a rougarou itself.
While 1935 is the latest date I can affirm, and all the 19th and early 20th works I've read don't involved infection, a 19th century date isn't out of the question. Varney the Vampire, confused as it was, states early on that a vampire bite won't turn a human into a vampire but later on victims are raised as vampires. With transmission by bite entering the realm of popular literature, and being very notable half a century after Varney in Dracula, infectious werewolfery could be a Romantic invention.