Many of today's rayon garments are labeled as washable. So yes while you can launder them at home there are some precautions you should take. Rayon can be extremely sensitive to water. Many dyes applied to rayon can bleed, causing a shading to occur upon contact with moisture. Manufacturers often add sizing chemicals to rayon to achieve a desired body or drape. Some sizing formulas are water-soluble and washing at home could distort the shape of the garment. Always use a short gentle cycle and cool water to be on the safe side.
If your rayon garment has a drycleaning care label you should follow those instructions. Some "dryclean only" garments contain dyes and sizing treatments that can dissolve in water.
Never try to remove a stain from rayon without first pretesting the fabric. Wet a small piece of the garment on an unexposed seam with the stain removal agent you are going to use and allow it to air dry. If no dye or sizing change occurs, a small stain may be removed in this way. If the stain is large you may want to take the item to a professional to prevent any color shading or loss of sizing.