The relay on time does not have to be exact. I have bought some Uxcell vibration sensors from Amazon , i was thinking glue it to the buzzer to trigger relay or transistor when buzzer on and vibrate. But this sensor is unreliable , uncontrollable, stay on and off at any time. If there is a device activate by sound , it would be easier.I'd agree with @wayneh then. A 555 timer is probably the easiest solution if you need the relay to be active longer than the LED.
I suppose the other possibility would be tapping into the buzzer circuit, but it might be tricky to find the right spot in the circuit to tap into to get a clean, sustained signal. The signal at the speaker itself is almost certainly some sort of AC signal which would be trickier to convert into a relay control.
Do you want the relay to be on the exact same length of time as the buzzer, or is it ok for the relay timing to be a little different?