Is a man at fault if he prevents his wife from performing 'Umrah with her relatives, due to his financial inability to accompany her, and his refusal to be indebted to anyone, despite her relatives offering to pay for the 'Umrah expenses?

This offer made by your wife's sister's husband to your wife does not obligate her to accept it, nor does it make Umrah obligatory upon her. You are correct in not permitting her to travel without a mahram for a journey that is not obligatory upon her.