I can totally relate to not wanting to have to transport it. For many years I did the same, just rented my ECV each day at the parks, for a similar reason... I was terrified of loading on and off the buses with it. It took years for me to get over that fear, but I'm happy I finally did... I love being able to take it around my resort and to Downtown Disney. I'm very much looking forward to being able to take it to Universal with me this year, because the walk from the parking structure to the turnstiles is SUUUUUPER LONG.
Anyhow, I digress! I totally understand the desire to avoid the hassles that come with renting from off-property, and just using the ones available for rent in the parks each day. My biggest piece of advice here is to get there EARLY, as they can and do run out as the park fills up.
And yes, you can share. In my younger days, before my health worsened, I traveled a couple times with a friend with back problems. To save money, for the first day or two of our trips, while our bodies were still fresh, she and I would get just one ECV and pass it back and forth throughout the day. Usually by the third day, one or both of our conditions would be aggravated enough that we ended up each just getting our own for the rest of the trip.
Hope that helps!