In Virginia and I assume in most states anything 50cc and up needs tags and endorsement and insurance.
We just passed a new law here requiring all scooters (even 49cc) to be tagged with a plate issued through DMV. It begins next summer (2014). I do not believe insurance on the 49cc is required though. I am sure they did this so we could collect personal property tax.
Simple answer is to find out your states laws but I feel quite certain that being a 125cc MC in every state the Grom will be treated as a full size street bike. In stock form, the Grom is 50 state compliant with lights, seat height, tires etc.