Praise be to Allah.
It is not stipulated that the one who is entitled to receive zakaah should be an adult of sound mind or should be eating solid food; rather the condition for being eligible to receive zakaah is what is mentioned in the Qur’aan of being poor or needy. Allah says (interpretation of the meaning):
“As-Sadaqaat (here it means Zakaah) are only for the Fuqaraa’ (poor), and Al‑Masaakeen (the poor) …”
Based on that, if this nursing infant has no wealth or has no one to spend on him and cover his needs, it is permissible for him to be given zakaah, which may then be spent on hiring someone to breastfeed him or on what he needs of clothing or medical treatment and so on.
Ibn Qudaamah (may Allah have mercy on him) said: It is permissible to give zakaah to both old and young, whether he eats solid food or not. Ahmad said: It is permissible to give zakaah in order to pay the fee for breastfeeding a foundling, who comes under the heading of the poor. And it was narrated from him that it is not permissible to give it to anyone but one who is eating solid food, but the former view is more correct, because he [the infant] is poor, so it is permissible to give it to him just as in the case of one who is eating solid food, and because he needs zakaah to pay for someone to breastfeed him, clothe him and meet all his other needs. So he comes under the general meaning of the texts.
End quote from al-Mughni, 2/268
But if there is someone who can take care of his needs, then it is not permissible to give zakaah to him, because he is no longer in need because of their support. It says in the commentary on Qalyoobi wa ‘Umayrah (4/28): The one who is sponsored by a relative or a spouse is not poor, according to the more correct opinion, because he is not in need; this is like the one who can earn every day enough to suffice him. End quote.
The majority of scholars (may Allah have mercy on him) said it is permissible to give kafaarah (charity given in expiation) to a nursing infant, as is explained in the answer to question no. 66886.
And Allah knows best.