Hi luckygurl1227 and welcome to the family vacation critic message boards. You are far more adventurous than I am to even consider riding a scooter or moped!
When we stayed on Oahu a couple of years ago we stayed in Honolulu but did a day trip to the north shore. I don't remember seeing any scooters or mopeds and the road that we were travelling had fairly heavy, fast traffic. Hopefully someone else who has stayed in more rural areas will chime in with their opinion on where scooter riding might be appropriate and where one can be rented.
In the meantime, if you have any other questions about your upcoming family vacation please do return to the boards and we will do our best to answer.