Trace the fuel line back from the petcock to the carburetor. where the feul line goes into the carburetor there is a banjo fitting that can leak if the top or bottom gaskets are damaged when the fitting is removed. If it's not here, it could be a stuck float ot leaky bowl gasket. Either way pull the carb and clean it. You have to remove the air filter first, then the banjo fitting to get to the calmp bolt. It helps to remove the tin plate under the engine.
The carburetor is very simple and requires no adjustments. Usually the main jet is clogged when thay are left sitting with fuel in the carburetor, since ther is no drain on it. Ironically, the DelLortto copy of the carburetor used on the Kinetic has a bowl drain.
Parts for the Vespa are available from www.vespasupershop.com. Talk to Fabio, who is the moped expert there.
Once you get' her running you'll have a great little bike that will climb any hill since it uses a variator. Before trying to start the engine the first time, give it a shot of oil in the sparkplug hole to lube the rings. As long as you have the sparkplug out, make sure you have spark when both switches (on the headlight and on the handle bar) are on.