Jordan Patch, the park's owner, said he has received calls and emails from people around the country and outside the United States, inquiring about where to travel to visit the park.
Animal Adventure Park is closed for the winter.
Patch told The New York Times he set up the livestream to allow curious residents in the area to monitor April's pregnancy on their own and stem the steady stream of phone calls and emails from Southern Tier residents curious about her progress.
But with the webcam logging nearly 80,000 simultaneous visitors as of Saturday afternoon, the idea has had the opposite effect for Patch.
"I'm answering a lot more emails now than I was before," he said. "It's absolutely overwhelming. We are inundated with messages and emails, media interviews and requests."
Giraffes have a gestation period of 15 months. Patch had initially forecast a due date for the calf in January or February, but now concedes that was incorrect, meaning April could give birth at any time.
The park will only keep the calf until April is done nursing, at which point arrangements will be made to transfer it to another facility to avoid incestuous mating.