Is there a service that can store and mail recommendation letters to schools? I am currently applying to graduate schools, all of which request 3 letters of recommendation.

Schools currently accept letters submitted by a recommender on an online service, or sent through postal mail. 

Since the recommendation letters that are submitted on my behalf are similar for each school, my recommenders have to go through a sisyphic process:


Get the email from the school
Click the link
Find the recommendation letter file
Upload the file
Submit
Rinse and repeat for each school


tl;dr

What I hope to discover is a service where my recommenders can upload their letter ONCE, and then have that service dispense the letters to all the schools, instead of having the recommenders themselves do it for each school separately. 

Does such a thing exist?
Many schools offer such a service. For example, UC Berkeley


  The Letter Service will send letters to educational institutions to support your admission to graduate and professional school. We will also send letters to educational institutions for the purpose of supporting your applications for academic/educational and research employment, i.e., teaching positions.


You should check with your school/department to see what is offered.