Strollers are easy... Especially to & from RP along the walking path. We've extended many days longer than we would've been able by giving our boy's legs a rest.
We also bring a backpack w/ water, etc. and for rides where bags are not allowed, they do provide lockers. The lockers are *not* big however... They'll fit a 'daypack' sized bag, but not if it's stuffed to capacity.
My park backpack is a North Face Jester Daypack, and it's dimensions are listed as 18.5 x 13.5 x 6.5 inches so if you've got something in that general size, you should be OK.
It does take some maneuvering & pushing to get the bag in & out of the locker, but when its packed with water, a few snacks, towel, a change of clothes for the boy, sunscreen, etc. it'll fit fine... If he wins a scuba Bart Simpson @ the midway right before we go on MIB, and he & the wife put Bart in my bag, then I'm going to be in for a struggle with the locker.