Most schools provide a letter storage service via their career center or some other resource. There are also online letter holding services such as Interfolio. You can ask your letter writers to send the letter directly to the service or upload it whenever they are ready. The letters must be signed and should have an official letterhead. Acceptable letters are then the letters uploaded into the service. You will be able use these letters later on when you actually apply to medical school. These services are secure and they will not allow you to read the letter beforehand.
These services are helpful because you can upload letters anytime and they are safely stored. Then you can use the letters when you need them. This prevents any last minute scrambling to turn in letters for your application.