Check with your national postal service. In Canada I can print as large as 9.2 x 4.7 inches for a postcard that can be mailed to the US or internationally with a normal letter stamp. I can actually print even larger for cards that are only being mailed domestically.
For my last exchange I printed two cards on one 8x10 sheet of Ilford Art 300 which I selenium toned. Each 5x8 print was slightly trimmed to meet the standard. I used inkjet printer labels on the back of most of them although some of them I just wrote on and put in the mail. That's probably what I would do again in the future (no labels).
I am going to do a cyanotype for the upcoming round. My local art supply store had some pads of 4x6 blank postcards on watercolor paper, made by Fabriano.
I got a huge variety of cards in return. People have all kinds of ways of doing it and they all seem to make it through the mail quite nicely. It's a fantastic opportunity to see a lot of different printing styles.