It is permissible to get a bikini wax during the seven clean days, although it is best not to have it done within a day or two of mikveh, in order to avoid having any remnants of the wax on your body during immersion.
If you experience any bleeding from the waxing it would not invalidate your clean days, since only bleeding that comes from the uterus will render a woman niddah or invalidate the clean days. If you find a stain on your underwear but based on the location it is likely it came from an irritation caused by the waxing, you may attribute the stain to that source and it does not invalidate your clean days.
Please feel free to get back to us with any further questions.
B'Hatzlacha!