What you are asking about is a Guest Assistance Card. The GAC (we don't like to keep typing it all out on this board) says right on it that it is not meant to shorten or eliminate waits in line. What it is, is a tool to let the CMs know what sorts of assistance/accomidation someone with an invisible disability might need. That might be a place to wait where you could get up and move around rather than staying in one place. They don't give GACS that limit your wait to a specific amount of time and a note from your doctor saying you need short waits would not make any difference.
If you think a GAC would be helpful, go to Guest Services in any of the parks and explain your needs (but, be aware, saying "I need to avoid wating in line" won't get you anywhere. They need to know what sorts of needs you have so that they can try to meet them).
If you have a limit on standing in one place, you should be aware that for most attractions, the line is continually making small motion forwards. The only attractions where you would be standing in one place are usually the shows that load a large people into a theater at one time.
In your case, you might be best off renting a wheelchair, so that you can have a place to sit when you need to sit and you can stand and push it when you have reached your sitting limit.
Depending on just when you are going, you may find that the waits in line are not anywhere near as long as you expect.
During peak times (like holidays, the height of Spring Break, during much of the summer), the waits may be long for some attratction. But, even at those times, if you can go to the parks early in the day (like at opening), you can get on many attractions without more than a 15 minute wait. My DH and I went last July (a busy time) to MK for park opening. By 11:30, we had been on Snow White (twice), Pooh, Peter Pan, Haunted Mansion, Pirates and the Riverboat without more than a 15 minute wait for anything. We walked right on to some of the earlier things we visited without waiting at all.
And, often things that are very busy at one point of the day (like Spaceshipe Earth at Epcot is busy early on because it's the first thing people see when they get to the park). By mid-afternoon, it's often a less than 10 minute wait.
What would be the most helpful to you probably would be to use FastPass as much possible. Fastpass is a free service available at some rides/attractions where you put your park pass into a fastpass machine and a fastpass comes out with a time printed on it to return. When you return, you will be able to go on the attraction - usually with a 10 minute or less wait.
