Is it permissible for a girl who works in a beauty salon, and has no other income, to borrow from her father to perform 'Umrah, or should she rely on her own income?

Working in a women's beauty salon is not inherently forbidden. Rather, it becomes forbidden if the worker performs prohibited actions such as plucking eyebrows (al-nams), hair extensions (wasl al-sha'r), looking at 'awrat (private parts), or beautifying immodest women, because the Prophet (peace be upon him) cursed those who do such things. If her work is limited to permissible adornments according to Sharia, then it is permissible and lawful, and she may perform Hajj and Umrah and give charity from its earnings. In fact, it might even be desirable to meet the needs of upright women. However, if the earnings come from forbidden acts, then it is not permissible, and one must dispose of it by spending it on charitable causes.