Everything you need to know about Guest Assistance Cards (GAC) is in post 6 of the disABILITIES FAQs thread. You can find that thread near the top of this board or follow the link in my signature.
You don't need a doctor's letter to get a GAC, although some people are more comfortable asking with a letter (CMs will almost always refuse to look at it).
You may find though, that a GAC really won't be as helpful as you think, so there are some things/limitations you need to be aware of for planning.
There is a GAC that allows walking guests to use the wheelchair entrances/access, which bypass stairs and allow boarding at a place where moving walkways can be slowed or stopped. Using the wheelchair entrance or a GAC in general does not shorten the distance walked.
Many lines are accessible all the way to the boarding area, so guests with wheelchairs and
ECVs use the same line as everyone else. For those attractions where the boarding or unload area is not accessible, the point where guests who need the accessible entrance are pulled off is usually close to the regular boarding area.
In most cases, the only way to be assured of a place to sit while you wait is to bring a seat with you, in the form of a wheelchair,
ECV or rollator (walker with a seat). Most attractions do not have a place to sit while waiting, even with a GAC. Even if they do (a few shows) there is no guarantee that the seats will be empty when you arrive and CMs will not make any guest get up so another guest can sit. Some attractions have wheelchairs at the attraction that they can lend for going in the line - the attractions that have them are usually those where the line is wheelchair accessible, but not ECV accessible.
There are also attractions where all guests stand for a period of time - usually during a preshow, as they 'gather' guests in one spot for the next show, or as guests are waiting in one area for the guests currently viewing the show to see it and clear the theater. Page 2 of the disABILITIES FAQs thread has a list of those attractions.
The last thing to think about is the distance walked outside of lines/attractions. The average guest walks about 6 miles per day at WDW - much of it getting from place to place or waiting in lines for other things like counter service restaurant ordering where a GAC does not apply.
These things are why Disney recommends a wheelchair or ECV for guests who are concerned about their stamina or ability to walk/stand in lines.