I do not think so. You can drop letters and cards in the mailboxes throughout the park - they are delivered to USPS in the early morning (postmark will be Anaheim, not Disneyland). The closest post office is half a mile away (1180 W Ball Rd, 714-533-8182).
If you're asking about mailing out purchases made at Disneyland, I think you can. I don't know what the fees are, but I think this is an option similar to having your purchases sent up to your room at a Disney hotel.
Maybe someone else will have more information on it.