No, in australia no vehicle at any age needs to be inspected yearly, or even regularly.
If the police stop you for a driving infringement, they may look at your bike and give you a 'Yellow Sticker' if they deem your bike to be in need of mechanical work to be classified roadworthy. Then you must get it repaired and take it to the department to get it approved and the sticker removed.
However, that would only happen if your scooter is unroadworthy and in a dangerous condition.
The only thing you must do is make sure you pay the scooter registration and clearly display the tag they give you, which is changed every-time you pay the registration. Registration can be paid 3 monthly, 6 monthly or yearly, so make sure the police have your correct address as they will send out the renewal notice to you. If you don't pay this renewal, then you have to take it to the department to have it checked before you can re-licence it, so make sure you pay the renewal. You do have 4 weeks grace if your a little late, but it is best to pay them on time.