I would recommend renting a stroller from Orlando strollers, magic strollers, or apple scooters (they have a special deal right now). The strollers you get from these companies are going to be much more comfortable than an umbrella stroller or the ones you can get in the park.
As for your concern of bringing the stroller on the bus at the beginning or end of the day, these are the times we are happiest we have a stroller with us.....especially after a day in the parks it is nice to have a stroller to let the kids sit and relax while waiting for transportation at the park or getting from the buses back to your hotel room. Much easier to do this with a stroller than to have to carry children at the end of the day.
Also, considering the amount of walking you'll do each day, it may make sense to get a double stroller. (You know your 6 year old best, and will know if he/she needs it)
We have used off-site rentals for the past few years and have always been happy with the delivery, services, and products. The strollers you can rent (city mini single and double) are very easy to fold and unfold for the bus rides.
Good luck with your decision.