Yes—a lens can prevent the camera from releasing the shutter properly even though the shutter itself is in the camera body, not in the lens. Since your D5000 can shoot with no lens attached, that suggests the shutter mechanism is probably working.

The more likely issue is with the mounted lens or the communication between the lens and camera. The best next step is to try a different compatible lens on the D5000. If another lens works normally, your original lens is likely the problem. If the same error happens with multiple lenses, the camera body or lens mount contacts may need service.

Without the exact error code, no one can diagnose it confidently online. A local camera technician is a sensible next step and may be more economical than sending it to an official service center right away.