Modern Vespa

My son recently bought the above scooter with only 5k miles on it from an older person who didn't use it much.
Shortly after purchasing it my son was riding down the road and it stopped, my son stated that it felt like it ran out of gas. we couldn't re-start so we had to push it home and during the last 2-3 weeks i have been trying to figure out whats wrong with it.
The battery spins the motor over just fine.
It has a full tank of gas.
If I take the fuel line off the carb and spin the starter I get a jet of fuel come out the line.
I have taken the carb apart, soaked the jets in cleaner, made sure I could see through all the drillings and re-assembled.
If I remove the plug, put it into the cap, hold against the head, I don't see any spark when I spin the starter.
Interestingly I would have expected the spark plug to have been wet when removed after repeated attempts to start but it was not.

I have tried a different spark plug.
I have replaced the engine kill switch.

What should I try next?
The previous owner did say at one time the fuel vacuum switch failed.

If you don't have spark, you probably have a bad CDI. Read the full description in the following link. I believe this is the correct part for your scooter. Call scooterpartsco to be sure. And, if so, order it.

https://scooterpartsco.com/ricambi/021-vespa/cdi-coil-for-vespa-lx-50-piaggio-fly-50-2v-unrestricted

Frank,
Thank you for your input, have you also experienced this issue an did it fix it.
You are based very close to me except I'm a little closer to Detroit.
Do you have a favorite workshop / repair center in the area?
Robert

My 2006 Vespa LX 50 experienced the same scenario as described in the description for the CDI, where it died while riding, but after sitting for a bit would eventually restart. The scooter was 11 years old with 660 miles when I bought it, so fits the scenario of an early Vespa 50cc that's been in storage for a while. Replacing the CDI fixed it permanently, and it's been flawless for over 2 years and 3,000 miles.

But now that I'm thinking about it, I recall some people having issues with the spark plug wire not seated in the CDI all the way. Pull back the boot, remove the plug wire, then push the plug wire back in until it's firmly seated and reseat the boot. If that doesn't fix it I would suggest a new CDI.

I've been doing all my own work, so I'm not aware of any shops in the area that I could recommend.