Set the multimeter so it is on 'continuity' - it will buzz when you touch the probes together.
Take the battery out of the 'vintage buzzer'.
Test for continuity between both 1/2 of the inside of the case.
Check for continuity between the inside of the 1/2 with the button to the cantilever spring (strip of metal riveted at one end).
Check for continuity between the coil spring and the inside of the case.
Finally check for continuity between the coil spring and the cantilever spring.
It's hard to see how it's put together from the pictures, is the buzzer part plastic mounted to isolate it from the case ?