elcome to disABILITIES!
Some of the answers to your question depends on what sorts of restrictions the child has. Some people with heart conditions have many specific restrictions; some have few.
You will notice on the park maps that some rides have a little red triangle icon next to them. Those are the rides that have health cautions. Some of them are bumpy, turbulent rides taht could be a problem for someone with a back problem (an example would be Star Tours). Some have sudden changes in motion or position that might be a problem for pregnant women or someone with a heart problem (examples would be Mission Space and Rock N Roller Coaster). Some might cause increased blood pressure or heart rate if the patient is scared (example might be Splash Mountain at the drop or Dinosaur). Many of the examples fit into multiple categories.
Anyway, you will be able to tell from the park maps which rides have health cautions. There are also signs at the entrance to the queue and before boarding.
So, the first thing you need to find out is if the doctor feels any types of rides are not a good idea.
Other things - if the child has problems with stamina, you can bring or
rent a stroller or wheelchair (I would not recommend renting a wheelchair in the parks for a small child, since the wheelchairs are mostly adult size). There is a lot of walking involved in the average WDW vacation; just the distance around WS is over 1 mile. Most of it is between attractions. If you will be using a stroller for the child, it is possible to get a Guest Assistance Card that would allow you to bring the stroller into lines and have it treated as a wheelchair.
If the child needs to rest during the day, one option is to go back to the resort for a rest. We always found that to be a lot of work and by the time we got DD there, she was in no mood to rest. A good place we discovered was First Aid. There is a First Aid station in each park. The do provide first aid for injuries, but are also great if someone needs a quiet, cool, dim place to rest. They have individual cubicles with a cot, a chair and a small table.
Are there any rides you have questions about? What sorts of things is the child interested in?