Kick start wont engage on my vespa 150 - Peg Perego Vespa Scooter

there must be holes in the muffler caused by rust - replace it with a new original one...
Vespa PX 125 Stock exhaust vs Sip Road Sport 2 0

stick the key (brown or blue cover) in the ignition lock and turn right until lights come on on the imstrument panel - if yes you have the right key and the battery is on. then push the starter button at the right handlebar. If the battery is good the engine turns on....

which model of Vespa is it?
- the older models have a manual choke (near fuel handle) which has to be pulled before starting.
- then also turn off the light before start

in front of the kickstart lever is a nut bolt which you must unscrew.
- but before you do that you need to mark the precise position on the lever axle with a permanent marker - after removing the bolt you can tear off the lever (use WD40 and wobble action)

Try turning the idle screw which protrudes from the air filter box anti clockwise once it's running. Or try taking your air filter off and drilling two holes (5mm + 7mm) in the little 'heart shaped' depression on the bottom of it.

I don't know much about Vespa scooters but from what you're describing, your valves need adjusting. If the valve is standing open as little as one thousandth of a inch, you won't have enough compression to start the engine at kick starter cranking speeds. It will start when you push the scooter and bump it off. This is because you get a higher piston speed allowing the engine to start before the compression has time to bleed off. Check the valve adjustment,especially the exhaust valve.