Son had hip surgery on both hips the summer between Freshman and Sophomore years in college. Teams were asking not only for the medical records but also to include the MRIs. He had them on a disk.
Besides/included with the letter Trust In Him posted that was sent to the top 300 players, the player will also need eye exam records sent in.
As time gets closer and the player is in the top ~200, someone will come and do a "whiz quiz" (collect a urine sample). The player doesn't know when it will be collected. Son's happened to be right before a game during his conference tournament. I think the player can refuse to do the urine test, but that could also eliminate him from the draft.
MRI's on arms could be requested before the draft. Son isn't a pitcher and had an MRI done on his non-throwing arm (Shoulder separated during a play, but he rolled over to pop it back in. Bad timing for that to happen, but it all worked out).