There is a thread on DIS about strollers......it sounds like a lot of people see other people leaving cameras and backpacks and etc in their strollers....there are also many people talking about their stuff being stolen, and some of the stolen stuff is really minor (water cups, baby diaper bags). Frankly, I would never leave my camera unattended...losing the expensive camera would be hard to take but losing the PHOTOS/VIDEO of our precious vacation would be absolutely devasting. THe tripod....well, I've only ever had the $10-15 tripods, so that I'd probably leave if I had a stroller, because it's loss would only be financial not emotional.
You might consider getting a bag that your camera will fit inside while you're on the rides. My understand is that the rides that get you wet have lockers where you can leave items while you're on the ride (and they are free while you're on the ride). If your plan is to video just certain things like parades, then you might consider paying for a locker for the day and leave your camera inside until shortly before the parade. Obviously this won't work if you want to video all the precious moments of the trip.....and you'll have to either plan your route to take you back to the locker area at the right time, or park your family on a curb to save parade watching space and make a dash back to pick up the camera.
I see a lot of people (both men and women) in photos people post on DIS carrying fanny packs and small backpacks, which one could presume has their camera amongst other essentials. In fact, I'm one of those people who never carries a purse in real life (I actually use a man's wallet in my jean pocket)....but I bought an Overland Donner bag from REI.com to use on this trip to carry my camera, the autograph books, water bottle and other essentials, since my jean pockets won't suffice this time.