I worked on a few postcard apps about 5 years ago, for my experiments the economics were very difficult unless a) you charged a lot b) of recurring revenue c) for a niche. I gave up even when I found a viable model because I didn't want to spent the next 5 years working on this, thought I could make a lifestyle business but I think to be successful here you have to move a lot of postcards and that takes marketing muscle and ensuring a high CLTV. Margins are good but absolute profit is tiny. Also support costs will be high, people will be emailing you about lost postcards (inevitable given the many ways they can be lost, and they will be pissed) when you stand to make 30 cents. Giving one away for free seemed to help "validate" that delivery has an existence proof for a given address, but is very expensive.

Anyway free info: the thing that seemed to have legs was an app that lets parents send monthly postcards of babies. (Recurring, high margin.) Grandparents and other relatives still have the ones my friends sent on their fridge. I think if you took this idea and executed well you could be successful, but you would need marketing capital or capital to get a partnership with a big brand.

Good luck!