Praise be to Allah.
If this orphan who is in their care has no wealth or anyone to spend on his needs, it is permissible for them to give him some of their zakaah, because he comes under the heading of the poor. Allah, may He be exalted, says (interpretation of the meaning):
“As-Sadaqaat (here it means Zakaah) are only for the Fuqaraa’ (poor)
The basic principle is that anyone on whom you are not obliged to spend, it is permissible for you to give zakaah to him. You are not obliged to spend on this orphan, so it is permissible for you to give zakaah to him.
In Saheeh al-Bukhaari (1466) and Saheeh Muslim (1466) it is narrated that Zaynab, the wife of ‘Abd-Allah ibn Mas‘ood (may Allah be pleased with her), asked the Prophet (blessings and peace of Allah be upon him): Is it acceptable for me to give charity (zakaah) to my husband and the orphans who are in my care? He said: “Yes, and you will have two rewards, the reward of (upholding ties of) kinship and the reward of giving charity.’”
Ibn Qudaamah (may Allah have mercy on him) said (2/271): If there is among his family someone on whom he is not obliged to spend, such as an orphan who is a non-relative, then the apparent meaning of Ahmad’s words is that it is not permissible to give his zakaah to him, because he would be benefitting by giving it to him because by doing so he is saving himself some expense. But the correct view, in sha Allah, is that it is permissible to give it to him, because he is included in the categories of people who are entitled to zakaah, and there is no text, scholarly consensus or sound analogy (qiyaas) to suggest that this is not allowed. It is not permissible to exclude him from the general meaning of the text without any evidence. End quote.
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) was asked:
Is it permissible for me to give the zakaah of my wealth to orphans whose legal shar‘i guardian is my father who is married to their mother?
He replied: If it was stipulated that your father should spend on these orphans who are in his care, and your father is doing that, then it is not permissible for you to give them zakaah, because they have no need of that as a result your father spending on them. But if they are staying with your father without it being stipulated that he should spend on them, and they did not have any wealth from their own father [i.e., they did not have any inheritance] then you may give them zakaah because they are entitled to it.
End quote from Fataawa Ibn ‘Uthaymeen (18/353).
And Allah knows best.