First of all, we would like to stress the fact that removing unwanted hair from the body is an act of fitrah. Referring to this, the Prophet (peace and blessings be upon him) is reported to have said: “The fitrah is five things: circumcision, trimming the moustache, cutting the nails, plucking the armpit hairs and shaving the pubic hairs.” (Reported by both Al-Bukhari and Muslim).
In Islam, we are certainly not allowed to expose ourselves before anyone except our lawful marriage partners (husband in the case of a woman, and wife in the case of a man).
So, if waxing the private parts is done by anyone other than the lawfully married partner, it shall be definitely considered as sinful and therefore haram; so never even contemplate such an activity.
All safe methods to remove unwanted hair from the body are permissible in Islam. In the past, Muslim jurists allowed the use of lime or An-Nawrah or some other depilatory agents; so in a similar fashion modern lotions (if they are safe for the skin) can be considered permissible. It is also permissible to remove hair by waxing, plucking, etc. But this should be done without exposing one’s private parts to another except one’s spouse.