Hello. The short answer is to ask the school what type of evidence they will accept. They should have an office (or maybe just one person) handling this.
The students I have worked with as a vocational counselor and evaluator over the past few years (okay, it's really been 3 decades, but admitting it makes me feel even older) typically provide the college with a letter from their doctor that includes the diagnosis, or diagnoses, limitations and recommendations.
Here is how a young lady I met for the first time last week went about it. She was about to be suspended from college for the second time in 2 years for low grades when she decided that maybe she really did need some accommodations.
Not too long ago she arranged to have a neuropsych eval done in order to provide the school with evidence that she needed accommodations for limitations resulting from hydrocephalus. The school has been very helpful and she was referred to us(the state VR agency) for assistance with putting the last couple of accommodations in place and some additional career counseling.
John