Adoption in Islam and Its Types
Question

If a person asks to adopt a child from the orphanage, is it permitted for those in charge to give him what he wants?
Praise be to Allah.Forbidden type of adoption
Adoption in Islam of children is of two types, forbidden and not forbidden.
The forbidden type means adopting a child in the sense that the child is considered to be the child of the adopting parent and subject to the rulings on children . This is not permitted. Allah nullified it in the Quran when He said (interpretation of the meaning):
“…nor has He made your adopted sons your real sons…” [al-Ahzab 33:4]
Permissible type of adoption
The kind which is prescribed and may be mustahabb means being kind towards the child and giving him a righteous religious upbringing and sound direction, teaching him that which will benefit him in this world and the next. 
To whom should the adopted child be given?
But it is not permitted to hand a child over except to one who is known to be trustworthy, religiously-committed and of good character, who will take care of the child’s interests. He should also be a local resident, so that he will not take the child away to a country where his presence may be a cause for his religious commitment being lost in the future.
If these conditions are met in the case of both the child and the adopting parent, then it is OK to hand over a foundling whose parentage is not known. May Allah preserve you.
And Allah knows best.