I would normally say rebuild yourself because its not complicated if you're just replacing parts. Since it's a project and not sure if motor been open before, no telling what you have inside that motor. Maybe it's just a bad crank. bad lower end or a bad install of the top end. But it could be more and you will need someone that knows what they are doing to make sure all the correct gears and other components related to that motor is being used. Also, adding a sidecar requires something more then a 125cc top end. For that motor, 150cc will work but a 166 or 177 top end will be better. Of course a P200 motor is ideal if you can find one at a decent price.
I suggest several options.
1. Start researching and rebuild yourself. If you think you can tackle a correct rebuild yourself and have questions, this group can help you along the way.
2. find someone local that has worked on vintage vespas for years to help you get the motor complete properly and in the future give you a piece of mind. Not necessary a mechanic but another vintage rider in the area can help a lot.
3. If you can't find anyone locally or don't want to rebuild yourself correctly, send the motor to Mercato. They have the best Vespa motor rebuild prices anywhere and Matt has been working on these things for decades.
http://www.scootermercato.com/Garage/Engine-Rebuilds
Good luck!