All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
We have already clarified in Fataawa 83795 and 81210 that it is not permissible for a woman to travel without a Mahram (i.e. a husband or a person whom one is permanently prohibited from marrying due to suckling, marital, or blood relations).
What we may advise you to do is to divide your vacation, in a way that you my allocate some of it for ‘Umrah and some other for travelling with your wife to your country and by doing so, you can combine between the two matters.
If this is not possible, then you may go for ‘Umrah and when you want to travel with your wife, you should take an emergency leave even if it is an unpaid leave for two or three days, during which you may travel with your wife and then return to work. If this is also not possible, then although the ‘Umrah is obligatory according to our view here in Islamweb, and that it is an immediate obligation according to the view of the Hanbali School of jurisprudence, yet the majority of the jurists are not of the view that it is an immediate obligation. Rather, many scholars are even of the view that ‘Umrah is not obligatory, and that it is only desirable and this is one of the two narrations from Imaam Ahmad and this is the view chosen by Ibn Taymiyyah . Therefore, you may act upon the opinion of the scholars who say that it is not an immediate obligation, and you may travel with your wife and then perform ‘Umrah at another time if this is possible.
Allaah Knows best.