I started getting post cards from Disney after I stayed on site for the second time. Both times we stayed at the poly but I didn't get any postcards until a month after we got back from the second trip. Maybe they didn't have the post card program after our first trip. I'm not sure. My friends stay at AS once a year and have yet to get a postcard. Maybe it's where you stay.
I do know that you never had to present the postcard, just know the code to get the discount. I haven't received a post card in a while so I'm not sure what discounts are out there.
Louisa - I'd love to get a post card too! I've stayed on-site twice (once at the Poly last November), but nothing yet. Generally you will be asked to produce the post card at check-in at your Disney resort in order to qualify for the discount.