Well, you need to make sure that all your LOR writers are willing to have your completed letters doen by May 15th.
If they all agree to do this and they all actually do it, then on May 15th, as soon as you open your AADSAS account and enter in the name of your LOR writers, you can print off the LOR matching forms. You would then give the LOR matching forms to you letter writers so that they can send the matching form and LOR together in the mail to AADSAS (probably should give an stamped and addressed envelope to them)
If, on the other hand, you have a service at your school, like a prehealth dept. or a career services that compiles your LORs for you, you would ask that all the LOR writers submit their LORs to the letter service by May 15th. Then you take the same Matching forms to the LOR service at your school and ask that your LORs go out with the matching forms to AADSAS.