I've looked into this, as I had the same idea as you. The laws vary considerably state-to-state. What consitutes a "moped", "scooter", and "motorcycle", and the specific licensing requirements for each are what you have to watch for. CA is the most restrictive that I am aware of. In most states, the magic cut-off number is 49cc (or 50cc) as being the maximum engine size, and frequently there is a top speed somewhere in the 30s. Many 49/50cc scooters are governed to stay under this maximum speed just for this purpose (though they can be modified to go faster, but their classification then changes).
In most states, you need a valid driver's license to drive even a 49cc scooter. The benefit of the smaller engine is that in most states you do not additionally need a motorcycle endorsement. In my state, as most, I cannot even drive a 49cc scooter since I cannot otherwise legally drive. The fact that I do not need a motorcycle endorsement is inconsequential.
There are a few states that don't require a license at all so long as the scooter meets the size engine and speed restrictions. Frequently to meet this requirement, the vehicle must also have pedals, thus making it a "moped" or "motorized bicycle" which has more lenient rules.
The only way to know for sure is check with your state's DMV website. This info is usually pretty easy to find, as it is frequently asked.
One more important thing: In many states, depending on your specific circumstances, you can get a provisional license, even if you do not meet the medical conditions for normal driving privileges. Such provisions often restrict driving hours, passengers, etc., and are considered on an individual basis in my states. Presumably, it would be easier to secure a provisional license if you were only riding a scooter, though I don't know this.
As a side note, my advice is to avoid Vespas unless money is no object. There is nothing wrong with the quality, but you pay a lot for the name "Vespa". If you opt to get as scooter, check out Genuine Scooters. Their scooters are widely considered as good or better than Vespa, but at about half the price. Ask around, you will not find a bad word about them. The local repair guy says the only thing he sees them for is routine cable adjustments - never seen one stop working for any reason.
Scooters are fun.
-B
