No, we've always had a car. They used to offer a shuttle for a specified amount per person to Universal. It was limited in time (I think it picked up at 6 p.m. to get back to SOG), and we were looking to book it for June. I guess they are either no longer or temporarily not offering the shuttle since it's not on their transportation page.
If you're not big on rideshare (we aren't), you might want to look into renting a car for 24 hours from Disney Car Care. We've gotten some prices that make it worth it and Car Care's shuttle will pick you up and drop you off anywhere on property. (for the pick up, you'd want to arrange a ride over ahead of time but for drop off, they simply ask where you want to go).
We pretty much rent through Costco. We are picking up a car at Car Care in June for the week with our rental house/SOG split stay. It was $350 cheaper to get it on Disney property than at the airport. Crazy!
I priced out Mears and some other places, and even with paying for a trip from the airport, will still save us $250. Worth the inconvenience and I figure lines for the rental are way better at Car Care than at the airport