We had strollers every year up until last year and my daughter had turned 6 while we were there. She runs hard all day at home, but in the park, just kept complaining and we kept carrying her. This year is the first we haven't had a stroller.
IF you buy a cheap stroller, be sure to get one with a canopy on it!! And be sure it's comfortable and your little one fits in it. My youngest is pretty small, so she fit into the umbrella stroller at 5 years old, but at nearly 7 years old, she is barely 45lbs if I remember correctly.
We rented the strollers last year, and even if you come/go from the park, you can turn in the stroller, and get a new one when you come back for free. (very handy). However, at $8 a day, that can add up to a nice chunk over 5 or more days, you may as well invest in a nice $40 stroller and sell it later at a garage sell or hand it down to a friend. (I am SO cheap and hate paying for something I don't get to keep, but will admit, not having the burden of a stroller around last year was nice too, no loading/unloading it from the car or shuttle. :s)
Every year we had a stroller, we just left it in the stroller parking area, even with some stuff in it, and it was never bothered. The only thing that we had happen was if we parked in an "unauthroized parking area", we would come out and have the stroller moved to it's rightful parking area. I wouldn't leave my camera or purse just sitting in it, but if you had a stroller with a pouch for hiding stuff, I wouldn't worry too much. For some reason, Disneyland just seems "safe" to me. (I know.. I know.. one day I'll have something ripped off and my dream will be shattered)