As the others mentioned, a Guest Assistance Card is not meant to bypass lines or necessarily to even shorten the wait. What it is meant to do is provide conditions (such as a quieter place to wait, a place out of the sun, etc) so that you can experience the ride/attraction with the least amount of stress.
You don't need a doctor's letter, but do need to be able to explain what your needs are.
Fastpass is available to anyone and does help to limit the wait in lines. You put your park ticket into the fastpass machine and a fastpass comes out with a time printed on it. At that time, you go back to the attraction, get in the fastpass line and your wait for boarding will generally be 10 minutes or less. You can send one person ahead to the fastpass machine; they can feed your whole party's tickets into the machine and get fastpasses.
Other ways to limit your wait include going to the parks early - even during busy times, you can often ride a lot of things in the first few hours after the parks open without long waits or using fastpass.
Also, look at the wait times for the standby lines before getting into line. Frequently, the waits are not as long as people expect, especially for rides that have fastpass since so many people are using fastpass. You will also find that some rides with long waits early in the morning are walk on later in the day. That's because people tend to go on the first ride they see - for example, Spaceship Earth at Epcot is frequently a 45 - 60 minute wait in the early morning, but by mid-afternoon, you can walk right on.
Each park has a "Status/Information Board" near the front of the park. The CMs there can be very helpful in deciding where to go and when for the shortest waits.
We recently got back from a visit (it was Spring break and very busy) with my nephew who is on medication for ADD. We did have a GAC for my DD who has multiple disabilities, but we used it only once or twice a day. And (or course) on the rides that she could not go on, we could use her GAC. My nephew did quite well. He had a few issues where he got bent out of shape and had meltdowns for rides/attractions he was not personally that interested in riding on. Interestingly, the ones that caused the most trouble were not the ones that had the longest wait - they were the ones he was personally least interested in.
