Yes! All resorts sell stamps at the front desk and you can either leave your postcards with a castmember or find one of the mailboxes within the resort. They aren't the blue mailboxes, though, so you may have to ask where it is.
There is an old-fashioned, AND REAL, mailbox on Main Street in the MK. I love mailing my postcards from there. It's on a lamppost (if I remember correctly) and it's green and not too big.
Have fun!