Push male half of grommet through hole and rest on anvil; add female half, set mandrel in place, and hammer.
For both versions, make a 1/4-inch hole on box ends. Place ribbon spools inside, then slide a 1/4-inch dowel into one end, through the spools, and out the other. A thumbtack on each dowel end keeps it from shimmying into the box.
Utility knife or grommets
Perhaps it's time to graduate from the shopping-bag school of organization. Ribbons will stay untangled and ready to use in this easily made box (a shoebox will do). Use a utility knife to make slits. Or use grommets; you can find them, and the tool to install them, at hardware stores.