Stamina, fatigue are serious issues. However; please do not suggest a wheel chair as children just lost a family member to the same disease and parent can not place the children in more fear/stress at this time.
I'm not sure why a wheelchair would place the children in more fear or stress and, even though you have written it is not an option, I think it would still be a good thing to investigate.
If the person using it would be benefitted by having their stamina and fatigue aleviated by using a wheelchair, it would benefit the whole family. Even with a GAC, there is a lot of walking involved in a trip to WDW and a wheelchair or
ECV (for someone over the age of 18 yrs) is a way to preserve energy for the fun things instead of 'spending' it all just getting around. WDW is very large and estimates are that the average person walks between 3 and 9 miles per day at WDW. Just to walk around World Showcase in Epcot is over a mile, and it's a similar distance to get into the park!
IMHO, it would be potentially more stressful for the children to see the person having problems with fatigue, stamina or pain from trying to walk around than it would be to see the person relaxed in a wheelchair. One way to explain it that works even for pretty young children is to be very matter of fact and equate a wheelchair with something they are familiar with like glasses = "You know Aunt Betty (or who ever) has glasses. That is to help her see. The wheelchair is to help ________ with getting around without getting too tired." If the adults see the wheelchair as scary and stressful, the kids will too. If the adults treat it as nothing to be concerned about, most kids will too.
If you are talking about a child, the WDW strollers are large enough for up to a small 12 yr old; some people get a double stroller to have more room for stretching out. As was already mentioned, you can get a GAC (Guest Assistance Card) that will allow a stroller to be used as a wheelchair.
Another benefit is that the wheelchair or stroller will be a 'buffer' to limit how close others get to the person using it, which should help limit exposure to others.
WDW will accomodate keeping someone out of the sun, but most of the queues are shaded at least to some extent, so this might not help that much. Also, you will still have to worry about the sun when going between attractions.
I think you/they need talk to the doctors regarding what to do to limit exposure, other than the things listed in this thread. While you do want to limit exposure to people with cold like symptoms, many of the infections that people with severly compromised immune systems get are from germs that they carry around in their own bodies. No matter how careful you are to avoid 'outside' germs, you can't avoid the ones from the germs 'inside'. So they will need to be prepared for the person possibly getting ill while at WDW; things like bringing information from the doctor about the medical history; getting suggestions from the doctor about who/where to go if the person becomes ill.
Another thing that would be helpful (potentially more helpful even than a GAC) is to know when different places are busy. Even in busy times like Easter, it's possible to go to all the Fantasyland attractions with very short waits if you can get there early after the parks open. Then, by the time the park starts getting busy, you can leave and go back to the resort for a rest during the busier and hotter time of the day. Things like Tour Guide Mike and some of the park touring books can help you figure out busier times and days so you can avoid them. Some of the advantages of doing that is you will be able to see more in a shorter amount of time, an easier time getting around and with less exposure to other people, since the crowds are lower.
The person may also have a time of day that is better for them. Some have lots of energy early in the day and then get more tired as the day goes on. Some people take a while to get going in the morning, but improve as they loosen up/ wake up. Working against the person's normal 'flow' will take more energy.
The last thing I can think of is to plan your days according to the weather - if it's a really hot day, that will zap energy, so you want to plan for an earlier day or plan to come back to the park for the evening when the sun will be down and it will be cooler.