Is there a program that will address a mass amount of post cards through my printer?

You should be able to use whatever you’re using to generate the labels (maybe MS Word) to mail merge a letter that only has the address for the postcard.

That is, one of the features of word is to generate form letters with different addresses/names/database info. So you can use that to create a postcard sized letter that contains only the address.

The catch here is that the postcards must be printable by your printer… but that’s not a big catch.