All perfect praise be to Allah The Lord of The Worlds, and may His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
Zakat (Alms) is an obligatory act of worship that has special conditions. These include making the Zakat into the possession of the poor and needy, and paying it immediately once it reaches Nissab (Minimum amount liable for Zakat) and one full lunar year lapses over its possession. Al-Imam An-Nawawi (May Allah have mercy on him) said, "Zakat must be paid immediately once the giver (Person on whom Zakat is due) has the money and there are eligible recipients." {Minhaj At-Talibeen, 72}. However, building an orphanage for the benefit of the poor doesn`t meet these conditions.
The description "Orphan" doesn`t make him/her eligible to receive Zakat, because an orphan could be rich. Actually, the Holy Quran has fixed eight categories of recipients: "Alms are for the poor and the needy, and those employed to administer the (funds); for those whose hearts have been (recently) reconciled (to Truth); for those in bondage and in debt; in the cause of God; and for the wayfarer: (thus is it) ordained by God, and God is full of knowledge and wisdom."{At-Tawbah, 72}.In addition, Narrated Ziyad ibn al-Harith as-Suda'i:I came to the Messenger of Allah (PBUH) and swore allegiance to him, and after telling a long story, he said: Then a man came to him and said: Give me some of the sadaqah (alms). The Messenger of Allah (PBUH) said: Allah is not pleased with a Prophet's or anyone else's decision about sadaqat till He has given a decision about them Himself. He has divided those entitled to them into eight categories, so if you come within those categories, I shall give you what you desire.{Related in Sunnan Abi-Dawoud}.
The poor and needy-be they orphans or not-are amongst the eight categories eligible to receive Zakat. Therefore, it is permissible to pay it to the orphans who are poor; provided that it is made into their possession, as indicated by the (لام) in the above verse. Al-Biqa`i said, "After mentioning the first four categories eligible to receive Zakah (the poor and the needy, and those employed to administer the (funds); for those whose hearts have been (recently) reconciled (to Truth), and that it should be made into their position, Allah, The Almighty, mentioned the other categories which are to be given Zakah to meet their particular needs: freeing a slave, settling a debt, fighting in the cause of Allah (Jihad) and for the wayfarer to reach his home, and this was indicated by (في) in the same verse."{Nathm Ad-Durar, 505/8}.
In conclusion, it isn`t permissible to pay the Zakat-available at the aforementioned charitable organization-for building an orphanage. However, it is permissible to use charities for that end. And Allah knows best.