Is it permissible for him to give zakaah to his sister so that she can get married, as her father has died but her mother is still living?

Firstly:

If a woman needs money to get married, then that should come from her own wealth, if she has wealth. Otherwise it should be paid by the one who is obliged to spend on her. That is because upholding chastity is one of the things that is included in spending on maintenance.

If your mother is well off – meaning that she has wealth surplus to her needs – then she must spend on her daughter.

Ibn Qudaamah said in al-Mughni (8/212):

She must spend on her child if he does not have a father. This is the view of Abu Haneefah and ash-Shaafa‘i. End quote.

If the mother is well off, it is permissible for you to give your sister zakaah if she is in need, because spending on your sister is not obligatory upon you in that case.

Secondly:

If the mother is poor, she is not obliged to spend on her daughter, and her daughter’s maintenance becomes the responsibility of her brothers who can afford it.

A brother is only obliged to spend on his sister if he is well off and she is poor, and he would inherit from her if she died, because father and grandfather are both deceased and she has no son.

Ibn Qudaamah (may Allah have mercy on him) said in al-Mughni (8/169): In order for it to be obligatory to spend [on siblings], there are three conditions, the first of which is that they should be poor and have no wealth or income that would make them independent of means and thus have no need for someone else to spend on them.

If they are well off, because they own wealth or have an income, then there is no need to spend on them.

The second is that the one who is obliged to spend on them should have enough to do so that is surplus to what he needs for his own maintenance, whether that is from his own wealth or from his earnings. In the case of one who has no surplus, nothing is required of him, because of the report narrated by Jaabir, according to which the Messenger of Allah (blessings and peace of Allah be upon him) said: “If one of you is poor, let him start with himself. Then if there is anything left over, let him spend on his dependents, and if there is anything left over after that, let him spend on his relatives.”

The third condition is that the one who spends would inherit from the recipient, because Allah, may He be exalted, says (interpretation of the meaning):

“And upon the [father's] heir is [a duty] like that [of the father]”

[al-Baqarah 2:233].

Moreover, there are ties of kinship between those who may inherit from one another, which dictates that the heir is more entitled to the wealth of the one who leaves it behind than anyone else. So it is appropriate for the heir to uphold the ties of kinship and spend on one from whom he would inherit than for anyone else to do so.

If he is not an heir, then it is not obligatory for him to spend on him. End quote.

Shaykh Ibn ‘Uthaymeen said in ash-Sharh al-Mumti‘ (13/503):

The guideline in our view is that it is stipulated that the one who spends should be an heir of the one on whom he spends, with the exception of ascendants  and descendants, for whom there is no stipulation that they would inherit from one another.

If you and your brothers are obliged to spend on your sister, it is not permissible to give her your zakaah unless she is in debt, in which case it is permissible to give her zakaah with to pay off her debt, because the one who spends on a relative is not obliged to pay off the debt of the one on whom he is spending.

It says in al-Mawsoo‘ah al-Fiqhiyyah (23/177): The Maalikis and Shaafi‘is, and Ibn Taymiyah among the Hanbalis, stated that what is not allowed is limited to the share of zakaah that is allocated to the poor and needy. But if he gives to his father or his son some of the share that is allocated to the workers who collect and distribute zakaah, slaves who have a contract of manumission, debtors and those fighting in Allah’s cause [on the basis that his father or son is included in one of those categories], there is nothing wrong with that.

And they said: If he is not obliged to spend on him, it is permissible to give him [some of his zakaah]. End quote.

And Allah knows best.