The primary purpose of de-restricting a modern Vespa scooter is to increase its power output to go faster (vintage Vespas have no restrictions). Restriction of a scooter engine is usually designed to create an arbitrary limit on how fast a scooter can be driven. This is regularly performed by the scooter manufacturer in response to regional regulations. However, aftermarket mechanics can circumvent these manufacturer limitations to realise the true mechanical output of a Vespa scooter for better driving enjoyment.
Identify where the limiters are on your scooter. Restrictions come in one of three forms. Some scooter manufacturers implement more than one restriction. These include the exhaust pipe, on the transmission/front variator and on the CDI (electronic controller).
Identify which Piaggio/Vespa engine you have. There are three different types: Pre-2000 Piaggio engine--No catalytic converter and Dellorto or Weber 12mm carb; 2000 through 2005 Piaggio/Vespa engine--Catalytic converter and 17.5mm Dellorto carb; 2005 onwards Piaggio/Vespa engine--Catalytic converter, 17.5mm carb and slightly lower exhaust port. De-restricting for pre and post-2000 engines is done in two parts. For the post-2000 engines, there is an additional step of adjusting carburettor jetting as well.
Prepare a clean work area and make sure all necessary tools and parts are nearby and ready for use.
Open up the Vespa engine case and locate the outer engine pulley. Undo the outer pulley only. You may need an air-power impact wrench or a variator locking tool to successfully stop the pulley from spinning while loosening. Locate the cylinder washer between the inner and outer transmission pulley when exposed. Remove the washer and reinstall the outer pulley. Nothing else needs to be removed. Secure the outer pulley and close up the engine case, making sure to secure all engine case nuts as needed to factory tightness. Use a torque wrench to be sure.
De-restrict the exhaust. The second restriction on a pre-2000 engine is a small pipe coming off the exhaust which interrupts the gas flow slightly. Remove the exhaust for the work to be done. Cut that small pipe off and weld up the holes created. This is a minor restrictor and will not solve all your problems if you don't remove the pulley washer described above. Reinstall the exhaust when completely finished.
Make jetting changes. Post year 2000 engines require carburettor jetting changes. No jetting change is required for pre year-2000 engines. Find the carburettor on your scooter. All post-cat Vespa models now come with a 17.5mm carburettor. The stock main jet is either a size 53 or a 56 (pre or post year 2005 model respectively). Take the old main jet our and replace it with a larger size main jet, two to three sizes bigger (56 or a 58).
Re-assemble the carburettor and close up the scooter.
Test the engine both in standing mode and in running mode.
This approach is not for the new rider or mechanically faint of heart. De-restricting a Vespa scooter requires some level of experience and mechanical knowledge working with scooter engines. Remember, not reassembling an engine correctly will likely result in a mechanical failure. At worst, it could result in serious harm and injury when riding. If you're not sure of your abilities to work on and repair Vespa engines, have a licensed dealer or shop do the work for you.
De-restricting a Vespa scooter in general is usually an attempt to circumvent a mechanical requirement put in place in the region the scooter was sold. That means you would be violating a local or regional ordinance driving a modified scooter around in that area. The results if caught could range from a fine/penalty to complete seizure of the vehicle by law enforcement. At a minimum, you will likely be forced to fix the scooter back to its original restricted condition to ride it again, once flagged. So consult with your local and state driving rules and laws before attempting a derestriction on your Vespa. It could save you a lot of time and money.