Praise be to Allah.
If zakaah has become obligatory for him, it is permissible for him to delegate a trustworthy person to distribute the zakaah, but it is better for him to distribute it himself and not to delegate it to someone else, so that he can be certain that it has been done.
It says in al-Insaaf (3/197): It is permissible to delegate someone to give zakaah, and it is valid to do so, but that is on condition that this person be trustworthy. This was stated by Imam Ahmad. [That person] should also be a Muslim, according to the correct view. End quote.
Al-Nawawi (may Allah have mercy on him) said in al-Majmoo‘ (6/138): He may delegate someone to dispose of the zakaah that he may distribute himself… Indeed it is permissible to delegate someone else to do that even though it is an act of worship, because it is like paying off debts, and because it may be necessary to appoint a delegate because the wealth is elsewhere and so on. … But distributing it himself is better than delegating someone else, and there is no difference of scholarly opinion on that score, because then he will be certain that it has been distributed, which is not the case when it is delegated to someone else. End quote.
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) was asked:
Is it permissible to delegate someone else to give zakaat al-fitr and zakaah on one’s wealth, or to accept it?
He replied: Yes, it is permissible to delegate someone else to give zakaat al-fitr, as is also the case with regard to zakaah of one’s wealth, but it is essential to make sure that zakaat al-fitr reaches the poor person before the Eid prayer, because the person giving it is acting on behalf of the one who is obliged to pay it. But if a poor man delegated someone’s neighbour to collect zakaat al-fitr on his behalf, and said, “Collect zakaat al-fitr for me from your neighbour,” then it may stay with the one who was delegated to do that, even if that is until after the Eid prayer, because if the one who is delegated to act on behalf of the poor man has acquired it, that is the same as if the poor man had acquired it himself. End quote from Majmoo‘ al-Fataawa, 18/310.
And Allah knows best.