Moped and scooter insurance are widely available. Motorbike insurance premiums are determined by the vehicle's engine capacity, so less powerful mopeds and motor scooters have lower insurance premiums than higher-end motorcycles.
Start by checking what kind of engine yours has. Mopeds typically have a very low engine capacity and are almost always less than 225cc, while most but not all scooters are.
225cc and under is the lightest class of motorcycles, and this category of moped or scooter will lower your insurance premiums more than standard motorcycle insurance.