Short answer: No.
Long answer: We HIGHLY recommend that you avoid any situation in which you might be tempted to dye a synthetic wig. Synthetic wigs are not like human hair and cannot be dyed with human hair dye.
Unlike human hair, synthetic hair is non-porous. The hair is extruded with the pigment included as part of the overall chemical composition of the material, and cannot be removed. You may be able to add rooting to non-rooted wigs with permanent fabric markers, but this is not a viable solution for all over the hair, as this usually fades or bleeds over time.
Some people try to dye hair with fabric dye, but we do not recommend this, either. This is NOT a natural fiber, like cotton. This is more akin to rayon or nylon and is VERY RESISTANT to dyes. It is made that way intentionally! (Remember: the color of a synthetic wig is part of the actual chemical composition of the hair, so it is very difficult to alter!)
The moral of the story: make sure you buy your synthetics in a color you can live with, because you cannot really alter it much. (Unless you’re a crazy scientific genius. In which case, there might be some money to be made, there . . . .)