Most camping chairs have been weather-proofed. As such, it’s generally not recommended to throw it in a washing machine, as this can remove the coating or shrink the material.
Instead, a simple rinse under warm water with a light soap (think Dr. Bronner’s Original) will usually do the trick. You can also use a toothbrush dipped in a water-vinegar solution for especially tough stains and caked-on dirt.