Is it permissible to postpone or pay in installments a debt incurred due to my oath to my younger sister to give her a certain amount of money, especially since I have not found the item upon which I made the oath, and I am in dire need of money? And do I owe expiation for forgetting to fulfill my vow to take my mother's item to the repairman?

If you swore to give your sister a sum of money if you found the item, and you did not find it, then nothing is binding upon you. However, if you swore that you would give her a sum of money if she found the item and she did find it, then if you do not do so, you have broken your oath and an expiation for the oath becomes obligatory upon you. An oath can be fulfilled at a later time, unless you intended immediate fulfillment.