Hi -
Haven't done it but this Vespa rental company in Venice Lido states that, "...the Italian Law minimum age for child passengers is 6 years old."
https://www.scooterrentvenice.com/faq---rentals.html
This one in Rome states that. "It’s forbidden to transport children under five years on two-wheel motor vehicles in Italy. For older children, the law establishes the principle that their size must allow them to sit in a stable and balanced manner, in the position determined by the vehicle’s equipment."
https://easybikerent.it/en/fai/
This one, also in Rome, states the same:
"Italian law strictly prohibits the transportation of children aged under 5 on two-wheeled vehicles. For children aged 5 and over, the law states simply that they must be large enough to sit in a stable, balanced way, in a position determined by the vehicle's equipment."
https://www.bicibaci.com/en/faq
So short answer, from what I'm seeing on various FAQ pages (you could look more of those up yourself for more rental companies) is that your 3.5 year-old is definitely too young, and it's sketchy enough for your 5 year-old that I wouldn't risk it. Heck, what with the hospitals overflowing COVID patients, I wouldn't risk one or both of you having an accident away from home with one of these scooters anyway, especially with young children along who depend on you, but that's just me. I will admit that if you've driven them in India, you likely know your way around Vespa operation but still....