Moyes said the club were waiting to see if the 28-year-old former Arsenal player would require surgery, but he is already planning to bring in a replacement.
"Vito has torn ligaments in his elbow and we're waiting to see whether it will require surgery. A minimum of three months," Moyes told a press conference.
"We definitely need to look for another goalkeeper because we only have a young keeper after Jordan Pickford."
Sunderland have lost 2-1 to Manchester City and Middlesbrough in their first two Premier League games and visit Southampton on Saturday.