We know, of course, that zoonoses – diseases transmitted from animals to humans – have existed about as long as humans and animals have co-existed, but what was the earliest documented case of a zoonosis? In other words, who first attributed a human disease to an animal source?
The answer depends on definition of "documented". There are documented cases of diseases that we now know transmit from other vertebrates to humans going as far back as 18th century BC, Babylonian codex Eshuna mentions “mad dogs” most likely infected by rabies. The Bible mentions an epidemic of (what appears to be) plague among the Philistines around 1320 BC, along with overpopulation of “mice” at the same time. See History of Zoonoses and Sapronoses for multiple other events.
"Documented" in the modern sense could not have happened before 1546, when Fracastro first proposed that infectious diseases are caused by germs ("seminaria morbi") in De Contagione et Contagiosis Morbis et Eorum Curatione, and then it took a while. The priority seems to go to Russian military doctor Stepan Andrievsky:"To prove that it was the same disease in animals and humans, he injected himself with the spores of anthrax taken from a sick animal in 1788 (or 1787). His diary helped him in the accurate disease description. He also originated the Russian name for anthrax: "Sibirskaya iazva" that is translated as "Siberian ulcer"... In all English sources the discovery of anthrax etiology is attributed to Robert Koch, and sometimes to Casimir Davaine & Aloys Pollender". Some English sources now do mention Andrievsky, although not yet Wikipedia. Koch's 1876 study is considered more conclusive than Andrievsky's, Davaine's or Pollender's, by then however other zoonoses (e.g. dermatomycosis) were documented.
It should be noted that although vaccination in the modern sense was only introduced by Jenner in 1796 (smallpox) more primitive inoculations originated in China and India as early as 16th century. According to Dietz and Heesterbeek:"infectious material from smallpox cases was transferred into the skin of susceptibles with the intention to induce lifelong immunity by a mild infection with a low case fatality. In 1721 this method was introduced from Turkey into England by the wife of the English Ambassador to Constantinople, Lady Mary Wortley Montague. It is not widely known that inoculation was even attempted against measles, plague and several diseases of animals (rinderpest, sheep pox, contagious bovine pleuropneumonia)".