Maybe paper dolls? “Paper” being a relative term; dolls of any medium in the flat paper doll style would work.
What about “color in an envelope” as a flat(ter) version of color in a box? (CiaB example here)
I think MistressJennie’s Scrappy Mail Fabric Swap is designed to be flat-mailable.
I feel like I had one more idea for this, but I’ve forgotten it…
ETA: …pretty sure it was prayer/friendship flags! Those are typically pretty flat, or can be designed to be, right?