Depending on how you define the word "zombie", Dawn, the ill-fated baby from Trainspotting (1996), might qualify. She isn't a zombie in the popularly accepted sense, because she doesn't eat anyone, and she is actually just a plain old dead baby, but during Mark Renton's heroin-withdrawal-instigated hallucination sequence, her corpse crawls across the ceiling to torment him.
Strictly speaking, it doesn't really matter whether Dawn is technically a zombie or not, because the movie Dead Alive! (1992) includes a baby who is definitely a zombie, by any definition of the word, and it predates Trainspotting by almost 4 years. Bizarrely, this movie was one of the earliest efforts from the man who would later go on to direct the smash hit The Lord of the Rings and The Hobbit series, Peter Jackson.
There are earlier contenders, however:
Burial Ground: The Nights of Terror (1981)
Redneck Zombies (1986-1987)
Pet Sematary (1989)
In most of these cases, everything hinges upon how one defines the words "zombie" and "baby": I think Redneck Zombies best fits both categories, but the zombie "baby" in Burial Ground is actually a toddler at the very least. On the other hand, the "zombie" baby in Pet Sematary might not be considered a true zombie - he's a toddler, and a young one at that, and he died and was reanimated, and he proceeded to kill everyone he met, but he didn't eat anyone, and he wasn't the typical stupid zombie (in fact, he was quite clever, in a monstrous and sadistic way).
In any case, the first movie to feature a zombie baby was possibly one of the above, although considering the decades of low budget zombie movies that have passed since the genre became popular, it is entirely possible that an earlier zombie baby movie exists somewhere. The most we can say with certainty is that the first zombie baby appeared onscreen no later than the mid-80's.