What is a "pedestrian entry gate" and why would it affect delivery vehicles. There's no parking at all, or no gate for the entrance to the parking lot? Or does the vehicle entrance have a working call box?

Perhaps whatever government agency is responsible for the inspection of rental housing will investigate, and order this repaired. While it's possible that the landlord has distributed a pass code to the police and fire department, it seems like the broken call box could create safety issues.