There are HD scooters that are travel scooters, but if you want one like you get at the parks it will probably not break apart. Also, keep in mind the larger the
scooter the heavy it is and the harder it is to get in and out of the car.
You may want to try a local dealer of scooters and see the biggest scooter that comes apart and that will fit in your car. You may be able to match the same kind when you rent. Take a measuring tape or ask for specs; so, you know how large the base of the scooter is so you can match size if you don't match the same kind of scooter. The Pride scooters are rented out a lot; so, try to see that kind when going to a local dealer. You can test drive the scooter at your local dealer, too. Make sure you like it. Depending on your group's size you can use your car not just your trunk to hold and put the scooter; so, it should fit in a large sedan. Let the dealer show you how it breaks apart and then you should try to do it, too. Put it together, as well. Fit it into the car; so, you know what to expect. Remember, you are responsible for charging the battery of the scooter every day. Try to get a scooter with a removable battery charge capability.
As far as getting a rental scooter delivered to your offsite resort all you can do is ask about it when you call the rental place. You can probably pick one up if you really want to have it at the offsite resort if they won't deliver there. Remember, you will have all your luggage and so forth with you and it may be very tough to fit the scooter in the car without two trips to Shades of Green.
I'm not even sure how rental companies handle delivery to Shades of Green; since, it's not an official Disney resort. Usually, the scooters are delivered to Disney resort's bell services (luggage area typically). You should call Shades of Green and see what they have to say if you have the scooter delivered there. Scooter rental companies will deliver early or the night beforehand sometimes; so, I don't see why you need to get the scooter off site unless you are planning to use it that day, but that's up to you.
It's a lit bit of work to get a scooter in and out of the car. It's also a challenge to use scooters on Disney transportation even monorails.