Are the sister and family considered sinful, in a religious sense, if the sister used the money she collected to buy an apartment for orphans (and it was conveyed to them that she is the mother of the orphans) to pay the installments of the apartment, instead of buying them supplies, clothes, or food, as some donors requested?

What the sister has done in terms of kindness is rewardable, for charity to a relative is superior. It is not necessary to inform the donors of her kinship to you because they gave charity to those in need, not to a specific person. It is permissible to spend the money on paying the apartment's price because it is among the needs of the recipient of charity. To alleviate any discomfort, the sister can inform the donors that she sends the charities "to the mother of orphans" for her to spend on her needs.