So voltage is at least 12.6?
I am not familiar with your scooter, but assuming we are sure the battery is good, if there are weak lights on the dash, agree it would seem either a relay, ignition switch or something like that is not working properly and may also cause the sound.
If you are not familiar, you can download manuals and wiring diagrams here:
https://manuals.wotmeworry.org.uk/Vespa/
Edit: agree with Tierney, get the battery load tested.
⚠️ Last edited by marret on UTC; edited 1 time