In RI its more than just a Doctor's note, there is quite a bit of paperwork, committees and meetings with a board specific to thes situations involved. These boards consist of medical doctors and experts in the field at hand. I'm sure every state is different, but heres how mine went.
I had to fill out the initial paperwork that I picked up at the Operator Control office portion of the DMV. My doctor filled out the basic questions on my conditions. They reviewed it, approved it, and sent him a huge questionaire with questions specific towards my condition (because I actually DO have one) and what his thoughts were. After he finished that paperwork and sent it in they reviewed it, approved it again, and asked for medical records. That is where I am at now.
Once I send them all of the medical records, they have the right to approve or deny me based on those facts. I was born with these conditions, and a surgery messed up my left eye even more, so I have legitimate reasoning. Im waiting for their approval to tint the front three windows and have the sticker placed on the car (which is never put in your possession here in RI the tint shop has it).
Thats how it works here. You are also allowed to have a tint exemption if you are carrying medicine that is neccessary to be with you at all times, and is photosensitive or heat sensitive. Again that is RI, MO could be way different.