What is the ruling on my sister's dealings with the Orphans' Fund in the manner mentioned, where the Fund purchases goods for her with an increase in exchange for installments, and the amount increases if she delays payment?

It is permissible to request to purchase goods or merchandise in installments with an increase in price, provided that there is no agreement to increase the amount if there is a delay in paying the installments, as this would be usury (riba). If the sister can deal with the fund in the aforementioned Sharia-compliant manner, without the condition of usurious increase, then it is permissible for her; otherwise, it is not. She must fear Allah, and He will make a way out for her and provide for her from where she does not expect.