I don't know of a way to ship packages from inside the parks, except for letters which can be put in a mailbox.
If you are staying at a resort onsite, check with their business center. I know Hard Rock and Portofino will ship packages for you for a fee, not sure about the other hotels. I would call and ask exactly how it works and the fees.
What we have done in the past is to pre-label and pack a couple of boxes packed flat. Then, you only have to drop them off at the mail center for pick-up once you've filled them up. I've never tried this at Universal, only at Disney, but I imagine it would work the same way.