I sure wish your school provided that service. It really makes it easy on the students. I would recommend you use a service then. Either that or you can ask the letter writers to make a number of copies and you could give them a stamped envelope. When you hear from a school (when you get a request for the secondary, that is when you send the LOR), you could ask them to send it directly to the school. It makes more work for them but it would be free. Make sure you have them sign the outside of the envelope stating they have not let you see the letter and have them sign the envelop flap. They must mail it AND make sure you don't put your address as the return address. You need to put their return address.
Since your school doesn't offer this service, I assume they don't have a standard form for all your LOR writers to fill out. Schools offer forms but that would mean your letter writers would have to fill out something new for each school. The schools also want you to sign a waiver stating that you haven't seen the letters. I think you need to talk this over with your people and see what they would prefer to do. If they don't mind filling out each school's form, then have them do that.
Are you applying to a lot of schools??? If so, then don't ask them to fill out a new one for every school. Try to get a hold of a standard form and do it that way.