Praise be to Allah.
One of the established principles of Islamic teaching is that everyone who acts on behalf of another person should act in his best interests. Therefore, what the supervisor of the institute must do is not use the buses belonging to that institute for anything other than the institute own interests.
When somebody is acting on behalf of another person, it is stipulated that he should act in his best interests.
End quote from al-Ashbaah wa’n-Nazaa’ir (p. 109)
Al-Qarraafi said: It should be understood that with regard to anyone who takes up the position of caliph and any position below that, it is not permissible for him to do anything except that which will be in the best interests of that position and ward off harm, because Allah, may He be exalted, says (interpretation of the meaning): “And come not near to the orphans property except to improve it” [al-Isra’ 17:34].
End quote from al-Furooq (4/76)
You did well by not allowing them to transport the mosque students to Taraaweeh prayer in the bus belonging to the Qur’an memorisation school, because this is not one of the interests that will benefit the women’s section.
If the bus is a waqf that was set up as such by the donor, then in such cases reference must be made to the conditions he made when giving the waqf. If he gave it as a waqf specifically to the women’s section , then it should not be used for any other purpose that is not connected to the women’s Qur’an memorisation section, even if it is charitable work.
But if he gave it as a waqf to the institute in general, then it may be used for anything having to do with the interests of this institute, including transporting its students to pray Taraweeh in other mosques.
It says in al-Mawsoo‘ah al-Fiqhiyyah (6/307):
If the one who gives a waqf stipulates conditions governing the waqf, then the majority of scholars are of the view that reference should be made to the conditions stipulated by the donor, because the conditions mentioned by the donor of a waqf are what govern the ways in which it is to benefit people; these conditions carry weight so long as they are not contrary to sharee‘ah.
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said:
It is obligatory to act in accordance with the conditions stipulated by the donor of the waqf… i.e., what was stipulated in terms of description, restrictions or the absence thereof, users and so on. Reference should not be made to the opinion of the supervisor regarding that; rather reference should be made to what was stipulated by the one who set up the waqf, and his conditions should be adhered to, unless they are contrary to sharee‘ah.
End quote from ash-Sharh al-Mumti‘ (11/12)
Shaykh Ibn Jibreen (may Allah have mercy on him) was asked:
Is it permissible to use cars belonging to charitable organisations for one’s personal business?
That is not permissible, because they are a waqf given to the organisation for the purpose of charitable work done by that organisation. So the worker does not have the right to use them for his own affairs, unless they were given to those who are in need, and the description of those to whom the waqf was given is applicable to him, whether that is on a permanent basis or in specific situations only.
End quote from the website of Shaykh Ibn Jibreen.
And Allah knows best.