Check out the disABILITIES FAQs thread near the top of this board (or you can use the link in my signature to get there).
Post #2 of that thread has information about renting wheelchairs,
ECVs and special needs strollers. You may be able to find a stroller the same as his to rent.
Post #6 of that thread has information about Guest Assistance Cards, where you can find out about using a stroller as a wheelchair. When doing this, you can bring the stroller with you into all shows for him to sit in during the show. You will also be able to bring it with in lines and keep him in it until boarding. It will be waiting for you when you get off the attraction.
During the time he is out of it, it will be sitting at the exit, which is a lot safer than just being parked in the stroller parking area. I can tell you from experience that most people getting off attractions are not seeing things like strollers when they get off. Thet are mostly focused on getting off and getting to the next attraction. I was really worried the first few times we left DD's >$8000 wheelchair, but soon saw that it was really not going to come to any harm.
Also check out the post on the FAQs thread about air travel. If your child can't walk for a short shopping trip, you will have trouble in the airport. You can gate check the stroller so that it is put in the baggage compartment last and taken out first. This is not a guarentee it won't be damaged, but it is much better than sending it throught with all the baggage. Also, in that thread, there is information about carrying a wheelchair onboard the plane. If your stroller meets the size and the plane has a closet, your stroller could be placed there. Depending on the size, it may even fit in the overhead baggage compartments.