How do I give a cardboard letter a distressed look?

You can paint it with chalk paint Sandy and then distress it with a light sanding. You only want to remove the paint not sand too deeply into the cardboard. You could use a darker color under a lighter color-two different coats of paint--sand lightly to let the darker color show. Finish your project with a coat of clear sealer or poly.

Maybe very very little paint on a dry sponge(2-3) colors that work well together, dab em allover separately then gently smudge them together with clean dry sponge n maybe dry paper towel!!??!

You can age it by painting it with wet tea and letting it dry. Make sure you don't apply it soaking wet or you may have a mess---

You can paint cardboard as long as it's sturdy. Thin cardboard will curl when painted. Apply a thin coat and use a piece of sand paper to distress the edges.

https://diyshowoff.com/2012/02/27/diy-aged-vintage-inspired-cardboard-craft-wall-letters/