Is it permissible to perform Tayammum for Fajr prayer if there is no fuel for heating, or gas, or electricity available, while fuel is available at very exorbitant prices and the weather is very cold, and fuel is purchased for cooking only out of necessity?

If you are unable to perform ghusl (ritual bath) after janabah (state of ritual impurity requiring ghusl) due to extremely cold water, fearing illness or worsening an existing one, and you cannot find warm water, or it is available at an exorbitant price due to circumstances, then it is permissible for you to perform tayammum (dry ablution). You are not required to repeat the prayer, based on the story of 'Amr ibn al-'As, who performed tayammum and led the Companions in prayer due to intense cold, and the Prophet, peace and blessings be upon him, approved of his action.