How to add a LED to a Multimeter for use it with the buzzer in continuity testing?
This may be tricky as you have no circuit schematic and have provided no evidence of investigation.



Figure 1. A piezo oscillator circuit. Source: Electronice Circuits.

Suggested approach:


Measure the voltage on each terminal of the piezo while silent.
Measure the voltages while beeping.


Even with only a multimeter you should be able to detect a difference.

When you've found the one that oscillates you'll need to monitor that with your LED circuit. This may require a high impedance input to avoid loading the piezo so much that it doesn't work.