Any reupholstery shop can do the work. When you take it apart, you will see how easy it is. Back comes off, base comes out and, if you have them, the end pieces come out as well.
We have done it twice in 11 years (the first fabric we chose was cheap and it picked up stains really easy).
While it was at the shop, we just moved two lawn chairs in and toughed it out for a week.
Old (usually dirty) Nissan Titan
Old (enhanced with "male pattern baldness") Sovereign
Young, lovely bride
Goofy dog, dismissive cat