Is there a service that can store and mail official transcripts from any school?
To my knowledge, there is no such service. Receiving universities want their copy to be certified correct by your university and sent in a sealed envelope such that you could not have easily tampered with it. No university is likely to trust anyone but the originating university to handle this process, so such a service has not been created. It would be very easy for some third party to allow you to create a transcript with whatever grades and courses you liked, and only through careful auditing could a university be able to trust that your records were properly transmitted. As such, it's easier, cheaper, and more reliable for the Registrar at each university to handle this on their own.