Vespa tours near San Luca

San Luca where?
There are at least 5 places and towns with that name all the way from Piemonte in the north to Calabria in the south.

I have driven a Vespa or a motorcycle thousand of miles in my youth in Italy. If you have no experience with motorcycles at home Italy is probably not the best place to learn. Riding a motorcycle or scooter in Italy is not for the feint of heart and is dangerous to the uninitiated. Highways are narrow and have no shoulders, therefore cars (and huge trucks) will pass you just inches from your side. If you still want to take the risk, then my recommendation is to have plenty of travel insurance.
If you don’t have a motorcycle drivers license (type A), and only have a car license (type B in the International Driving Permit) you can drive only up to 125cc. Vespas come in 50cc or 125cc. There are also larger ones, like 150cc, 200cc, and 300cc, but you need a motorcycle license for that. To mitigate the risk, being a novice, you should rent a 50cc, which has a top speed of 45 km/h (by law). That is about 30 mph. A Vespa 125cc can reach 95 to 100km/h (about 60 mph) but I can tell you that if you rent a 125cc without having driven a Vespa before, you are really taking your life in your hands. Only one person can ride in most 50cc models. A 125cc can have one passenger in the back. However to carry a passenger you need even more skill and experience.

I also put in thousands of miles on scooters from 50cc to 150cc back in the day, and after a couple decades away, I am returning to the fun this summer with a 450cc motorcycle -- I am less than 65 years old, and I have to tell you that I am being taken by surprise at how much slower my reactions are and how much less muscle I can bring to bear than I imagined. I was thinking that like a bicycle you never forget, but I am finding that things come at me quicker than I am yet comfortable with.

So, I concur with the above comment -- if you aren't very familiar with the vehicle you will be using then don't try to get the hang of it for the first time on roads and in traffic that you have never ridden before. And riding in a group makes things even shakier.

We want to tour country side rides with little/ no city driving due to our first experience (staying off busy roads)

I interpret that to mean you are not experienced 2 wheel motorized vehicle riders.
If that is the case I concur with Roberto, "Italy is probably not the best place to learn.". I would have omitted the 'probably'

Tourists driving cars are going to be focused on the scenery, and the locals may not realize in time that you are a tourist and versed with the local customs on how cars and scooters interact.

Make sure your insurance policy has medical, evacuation and repatriation coverage.

Did you change your original post? I missed the “Umbria” part when I read it last night, but see it now today. We were in Florence a couple of years ago, at an overlook above the city. Along came a Vespa tour, a string of 8-12 obvious tourists (American, I think), all one person per Vespa, most wearing helmets, but I noticed at least a couple had their helmet hanging on their Vespa, not on their head (a personal safety gripe of mine). Their guide pulled them over, and they all left their scooters idling. Some got off of the Vespas, some stayed seated, and it looked like they were maybe trying to figure out whether to go up to a church at the top of some steps. I suspect these were vintage Vespas, perhaps maintained well enough to use for guided tours, but their exhaust wasn't up to what I’d imagine were modern air pollution standards. Some even had smoke coming out of the tailpipe, and after a couple of minutes, the air in the area had gotten pretty foul. We left without having more time to enjoy the views.

The Vespa riders (maybe ages 30’s and 40’s?) were all laughing and clearly having fun. From the looks of them, I doubt that they were all really long-time motorized bikers, and probably not highly experienced with Italian roadways, but who knows? The guide made sure that they could safely get up along those roads above Florence, and had helmets, should they choose to actually wear them. The people at the back of the line of Vespas probably got the very worst of the exhaust pollution!

If you find an actual tour, I’m guessing they’ll make sure it’s safe for everybody. Maybe they’ll confirm whether you’ve, at least, ever been on a bicycle, then go from there. Does your hotel have any information about Vespa tours around that Umbrian San Luca?