I have a friend whose printing works used to have a mechanical stuffer - a device that folded letters, and inserted them into envelopes. This machine was ancient last time I saw it, don't know if it is still in use!

The expression seems to be in fairly common usage. M-W gives the following definitions:

Stuff vt (1a) : to fill by packing things in

Stuffer n (2): an enclosure (as a leaflet) inserted in an envelope in addition to a bill, statement, or notice.