It would seem that British film The Plague of the Zombies (1966) would be the first registered example of a zombie movie depicting the affliction as a plague or sickness. However, the origin of the zombies is that movie stays very much mystical in nature. To understand this duality, it is necessary to known the historical and cultural elements on which previous zombie films based themselves upon.
The invention of "zombies" as a concept (specifically zombies, not just undead-ness) is attributed to the Haitian culture. Eventually, foreigners assimilated the Haitian zombies to voodoo practices (which is a commun misunderstanding ; see here on Wikipedia along with relevant sources) and exported the concept. The first registered zombie movie ever was White Zombie (1933), produced by Hammer Films and starring cult actor Bela Lugosi. (Incidentally, Plague of the Zombies was also produced by Hammer Films.)
In Plague of the Zombies, the start of the movie depicts a plague sweeping through a rural city and killing the inhabitants one after the other. Eventually, the protagonists discover that victims from the plague have been zombified by the movie's villain through Voodoo spells. So in a way, Plague of the Zombies both is and isn't the first zombie movie to depict zombies as originating from a plague...
The first movie to clearly depict zombification as a sickness, though, is the obscure Hell of the Living Dead, where the infection stems for a chemical leak. If you specifically seek the first movie where zombification comes precisely from a virus, then it looks like Braindead (1992) might be it. The zombie plague comes here from a rabid Sumatran monkey's bite.
I may have missed an important movie, so feel free to point out a more correct solution if you have one.