I agree with most of what others have been warning against. The most costly items on a Vespa to replace are: frame, engine, and fork.

When inspecting a prospective purchase, check the frame first. If there are obvious creases along the floorboards, or traces of a violent blow on the sides or front, beware. The same applies for heavy rust under the bike (on T5s, by all means, lift the floormats for inspection, as water gets trapped under them). The Vespa's frame being it's sole chassis, it should be in good shape; otherwise the proverbial tricky handling could prove downright dangerous.

The fork is not really that expensive an item, but labor will be, and doing it yourself requires you to know what you're doing, and to be prepared to spend some time doing it. If the fork is bent, you should be able to sense it. But if the tires are funny or worn out, or the engine's loose, you might think something else is responsible for ill-handling, than the fork. So get on the scooter, let it off it's stand, and while you're watching the front wheel, turn the handlebars side to side. If the body moves up & down or sideways while doing this, or if the scooter seems to want to turn instead of drive straight while riding it, the fork is likely out of shape.  

The engine: when it's started, it should not make pinking noises, smoke excessively (when warm; the cold-started, choke enriched engine will smoke until warm, and the choke is cancelled) or leak. If you're looking for a restored scooter, it should rev nicely, gears should shift nicely, and not jump (when test-driving, don't hesitate to push all 4 gears to their end; if they're worn, they'll jump way before that)

In any case, my advice would be to buy the corresponding Haynes repair manual, and, if you have the time to spare, take the time to change all gaskets, cables, and ball-bearings, in engine & on frame. Also, always carry a spare clutch cable and tools with you, as it's the single item I've had to change on a regular and frequent basis...

My bikes:

1980 P200E,

1986 T5 (w/ Malossi 172cc, reedvalve 30 carb w/gasflowed crank, hydraulic clutch conversion, 5-speed gearbox, polyester flywheel, hydraulic disc brake conversion (front & back), Xenon headlights/LED lights everywhere else,...) 

1989 PX200E

2004 Triumph Speed Triple