We originally rented when we first brought my daughter. However, I did not like having to give the stroller up as we left the park because I ended up having to carry all the stuff while my DH had to hold her the entire bus ride home as she was passed out in his arms. After that first year, we brought our own stroller that we could collapse, but that had room underneath for the diaper bag, our purchases, snacks, etc. And the first day of each trip we bought a big balloon to tie to it so we could always find it.
We would still have to get her out when we got on the bus, but she (and the stuff) could stay in until the bus arrived at least (and sometimes we could just squish the stuff and it would fold up with it). And, when we got off the bus we could put her right back in. On the monorail, we just kept her in the stroller the whole time. And, there are certain boats, like the one we would take a lot to and from Wilderness Lodge, that allow you to keep the stroller as is and let the child sleep and the stuff stay. We didn't bring a super fancy one, but I think by the time I calculated how much it was to rent (I think it was $15/day) for 7 days, I just ended up buying like a $60 one at Target and bringing it. I gate checked it for the plane too so that she could ride in the stroller up to the gate and then immediately when we got off the plane, and then we didn't have to carry her through the airport.