So we hurried home, drove back to the chair, and popped it in the back of our pickup.
It was upholstered in vinyl, so I figured that it wouldn't hurt so see if I could get it cleaned up a bit.
I started out using a vinyl and leather car upholstery cleaner. It worked well, but I got faster results when I tried laundry soap and water.
Before long, it looked like a brand new chair!