Ease of use of SMD Led components
Conductive ink works wonders for this. A 0805 led and some conductive ink, flexing somewhat:



And a more complex one:



Bonus. No traditional soldering needed. The Ink is both glue and conductor.

The issue you may find is that a post office automatic sorter may bend the post card too much, or get caught on the led.

The biggest issue though, would be the power source. Any reasonable coin cell battery may be too thick, or not bend enough. And attaching it may not be easy.