I would bring a stroller, no question. DD4 and DS7 never ride in a stroller anymore except at Disney. It is mostly for DD4, who would always sit if she had the option but when DS7 needs a break, I know that I have a place to put him without slowing everyone else down. He knows that he has a spot to rest as well. It makes for less whining and a better time had by all.
Bring your own, a cheaper umbrella is always good. I have a double MacClaren folding stroller and I never, in many trips, had any problems with someone touching my stroller (except CM's who moved it to park, but that's not what I am refering too). This year I am thinking of renting. It is a expense but I tried one last visit and they push like a dream, especially with older stroller children.
It really is a lot of walking and as the vacation progresses everyone losing a little bit of stamina, Mom included.