The reason the teams want the medical and vision information early is so each club can have their own team doctors review the information. Each team has different opinions on injuries etc... for example a young player who had tommy John may give one team caution and another club not as much concern. Getting the information early saves the club time and money. If a certain team fails the players medical, then there is no reason to spend the time and money to go see him. If they pass the medical then they have the green light to continue evaluating the player.
Get any information requested to the teams as quick as possible, I can tell you from experience that a scout is not going to be as aggressive or fight for a player if they are unsure on their medical, signability, or character. The more information the scout and the team they work for has, the higher the comfort level will be when they start talking about the player and potentially drafting him.