Praise be to Allah.
It should be noted that the woman is not obliged to provide or furnish a home, rather that is the duty of the husband and it comes under the heading of maintenance that is his duty towards his wife. Similarly the mahr that is given to the wife is her property, and she may use it or part of it to prepare herself.
It says in al-Mawsoo’ah al-Fiqhiyyah (16/166): The wife’s trousseau: The majority of fuqaha’ are of the view that it is not obligatory for the wife to prepare her trousseau from her mahr or part of it. The husband has to prepare the house and everything that is needed for it to be a suitable home for them both. End quote.
If this girl has someone who spends on her maintenance, such as a father or relative then he is the one who is obliged to prepare her and give her to her husband. But if he is not able to do that, then the husband should do it and make that part of her mahr. See question no. 12506.
If the custom is for the wife to share in furnishing the house – such that she cannot get married otherwise, as is the case in some societies – and her wali (guardian) refuses to spend on her, or he cannot afford that, and she does not have any wealth with which to prepare herself, then it seems that it is permissible to give zakaah in that case, because the need for marriage is a genuine need, and in some cases it may be like the need for food, drink and shelter.
Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) said: If there was a person who can earn enough for food, drink and shelter, but he needs to get married and he cannot afford to get married, is it permissible to give him zakaah funds so that he can get married? The answer is yes, it is permissible to give him zakaah funds so that he can get married and give a full mahr. If it is said: How come it is permissible to give zakaah funds to a poor person so he can get married even if what is given to him is a large amount? We say: Because a person’s need for marriage is a real need, and in some cases it may be like his need for food and drink. Hence the scholars said: If a person is obliged to spend on someone, he should arrange that person’s marriage if he can afford that. So the father is obliged to arrange his son’s marriage if the son needs to get married but he cannot afford to do so. But I have heard of some fathers who have forgotten how they were when they were young and when the sons ask them for help in getting married, they say: “Get married from your own pocket.” This is not permissible and it is haraam if he able to arrange his marriage. His son will dispute with him on the Day of Resurrection if he did not help him to get married when he was able to. End quote from Fataawa Arkaan al-Islam, p. 440-441.
Based on this, there is nothing wrong with giving her some zakaah funds to help her with the wedding expenses. If you are afraid that she will spend this money on things that she does not need, then you can say to her: “I have such and such money for you; what do you want me to buy for you?” then you can buy her what she needs.
And Allaah knows best.