No, you will get nothing out of the transformer until the vibrator is running. So, lets test the vibrator.
You will want to use your trickle charger for this.
Remove the vibrator and hold it with pins up. Now put the larger pins toward you. Connect the negative to the large pin on your left and the positive to the small pin below it on your left. The vibrator should now buzz. If it buzzes, take a jumper and connect it to the other small pin. Touching the small pin on the left while it is buzzing with the jumper should draw a small spark or at least affect the frequency of the buzzing. If either of these tests fail, we can take it apart and probably get it working.
