OK let me get this straight:
If you are going to a Disney resort child care center, then I think it's easier to leave the stroller at your hotel and just walk your child to the care center. Or bring your stroller and fold it up, I think they will let you bring it. But that's for the child care at the resorts.
If you are at the PARKS, the baby care centers are not a place for ditching your child, they are for "care" - diapering, feeding, stuff like that (Which I think you know if you are a Disney Alumni, right?

)You must leave your stroller outside of them.
But.....if I were you, I would ditch the stroller at your hotel and carry your child to the child care/"babysitting" places that are at the resorts (forgot their names, but the ones like Simba's Clubhouse or the old Neverland Club). No real reason to bring a stroller.