I bring a backpack in the parks with me. It's not a large one, but it could fit 1 or 2 standard size textbooks in it.
I have never found any ride at Disney I couldn't bring the bag on with me. Most rides are tame, so there is room to just set the bag at your feet inside the ride vehicle.
On the more thrilling rides, like RnRC, EE, and TT, there is usually a pocket you can put a bag in. Depending on how much was in my backpack at the time, it didn't always fit in those. But I just looped the straps around my legs and braced my feet in front of me. I have never had an issue with a bag falling off that way, even with RnRC, which goes upside down.