Hi and thanks for the great question.
Technically speaking, no. Synthetic wigs are made of plastic with a very smooth surface so they can't absorb color like actual hair.
However, you can stain a synthetic wig. This technique only works going darker: you cannot lighten a synthetic wig. Using a fabric dye, like Rit™ brand, try mixing about 4 oz. product to a gallon of water and soak the wig in it until you achieve a desired darkening.
Wash it well with synthetic wig shampoo and don't forget to condition well too. Also, you can put the stain mixture in a spray bottle and mist the wig evenly until you achieve a desired color. This works great for layering stain tones.
I hope this helps.