I would consider renting scooters at a Disney resort. There's a resort onsite service to rent. You should do this is advance for your dates. Then, use Disney transportation with your Disney rented scooters to get back and forth from the parks.
You may also want to consider buying new lightweight scooters. You can get foldable one piece scooters that weigh 40-50 pounds. If you each take an end you should be able to lift those scooters in and out of your van.