I finally figured it out! This was a very strange problem that even internet research failed to diagnose. The scooter would essentially run out of gas when the fuel gauge showed just over or under 1/2 tank remaining. The first time it happened, I thought my fuel gauge failed and the bike was truly empty. This was not the case as I could see plenty of fuel in the tank. Nevertheless, once I filled up the tank again (by adding about 1/2 gal of fuel); the bike would start and run just fine. This problem happened several times and I finally started carrying extra fuel just to top it off. This was a decent solution to the problem of being stranded but was still annoying to worry about running out of gas at unexpected times. In traffic, this is not a lot of fun. After thinking through the problem, I recalled having a faulty fuel tap replaced prior to these issues. The mechanic who installed it assured me that this part would not do that. This mechanic worked at a Vespa Authorized repair shop in Daytona Beach so I initially took his word for it. I finally contacted Vespa of Orlando to see if they had a OEM fuel tap. They assured me that all of their parts come from the manufacturer and were all OEM. I decided to schedule an appointment to have the fuel tap replaced. Their mechanic has worked on Vespas exclusively for the last 7 years and has never hear of this problem but replaced the part anyway. This is what we found:
The fuel tap on the left in the picture above was the part removed and the fuel tap on the right is an OEM fuel tap for my Vespa. Apparently, the mechanic in Daytona installed a Chinese generic fuel tap that on the surface seemed to function ok. I do recall the mechanic having to make carb adjustments just after the installation because the bike “just wasn’t running right”. Sure enough, similar adjustments had to be made after the correct fuel tap was installed also. My only guess as to why this happened is because the vacuum required by the carburetor to operate the valve. I understand that there is a diaphragm and spring that close the valve when the bike is not operating to prevent fuel from flowing. The two parts must require different vacuum pressures and thus attribute to the fuel problems I experienced.
For anyone who finds this article though a search in the future, don’t accept Chinese parts on your Italian scooter. If you run out of gas like I did or have weird fuel problems after a fuel tap replacement then make sure the fuel tap was OEM before spending tons of money diagnosing fuel issues.