Part of the Arts and Craft movement, Mission style furniture was first designed in the late 1800s in California. Mission style furniture, including couches, is notable for its clean lines and simplicity compared to other furniture styles that featured ornate wood working. Finding a cover for a Mission style couch can prove difficult. You cannot use a standard couch cover, unless you are willing to conceal the wooden sides and back that give the couch its distinct style.
Call the furniture maker to ask if they have covers available for the couch cushion. Since a lot of Mission style furniture is built by small or custom furniture shops, they may be able to construct a cover for your couch or direct you to someone who can.
Take the seat cushion and any pillow cushions you have on the couch to a custom upholsterer and see if he will be willing to build a cover for them.
Sew your own cover for the cushion. If you cannot sew, you can try simply draping a large piece of fabric over the bottom cushion and tucking the ends into the sides, front and back of the couch. Find new pillow cushions to replace the old ones.