Welcome to disABILITIES.
We have been to WDW in September and it is a good time for 2 reasons; it is not so hot and there are not so many people there, so the lines will usually not be long. Most of the lines are shaded from the sun and some are even air conditioned (yes, outside lines!). If you can get to the parks early in the day, you should be able to ride a lot of things with minimal (15 minute or less) waits. And a lot of rides have fastpass, where you put your ticket into a mcahine and it prints up a pass with a return time. When your return time comes, you will wait only a few minutes in line.
The only people who get to go to the front of lines are celebrities and families who are at WDW with the Make A Wish (or similar) organization. They do have a Guest Assistance Card (GAC) that can be requested at Guest Services. You don't need a doctor's note; just go there with your child and explain your needs to the CM. It is printed on the card that it is not meant to give immediate access or shorten the wait in line. It will, though, give a more appropriate place to wait (like a quieter place or a place out of the sun). We get a GAC for our DD who is multiply disabled, but we usually don't need to use it more than once or twice a day, even over Spring break, when it is very busy. some of the rides that have long waits at certain times of the day have no wait laer (or earlier). We never needed it on our whole trip when we went in September because the lines were generally very short.
Another thing you might want to consider, since you say that your child tires easily, is a wheelchair. The distances involved in a trip to WDW are mcuh greater than most people imagine. Just going around the water at WS in Epcot is more than a mile! I've seen estimates that the average person walks between 3 and 8 miles a day at WDW. You might want to check if therre is an equipment lending "closet" in your area where you could borrow a wheelchair and bring it along with you. It's quite a ways from where your airplane lands at the airport to the baggage claim and there are no little golf carts to transport you. A wheelchair doesn't count toward your baggage limit.
