The Walt Disney Co.’s new streaming service may be the best thing going right now at the struggling entertainment giant, which is facing a once-in-a-generation crisis that has ravaged its movie, theme park and cruise businesses. But the company hasn’t spared the division that includes the fast-growing service—Disney+—from the frugality it has imposed on other businesses.

Employees across the sprawling streaming division have been told that any hiring now needs to be done strictly on an “as needed” basis, according to people familiar with the matter. Kevin Mayer, the head of the streaming division, is signing off on all new employees brought onto the team, the people said—part of a companywide policy that requires division heads to limit hiring to only those positions deemed essential. That has left dozens of open positions in the streaming unit unfilled, and executives have warned that isn’t likely to change until the downturn ebbs, according to one of these people.