Is it permissible to perform a free 'Umrah offered by charitable organizations for someone who is financially capable? And will the one who performs 'Umrah on behalf of a relative be fully rewarded for the 'Umrah, or is the reward limited to the one on whose behalf the 'Umrah was performed?

There is no objection to performing Umrah at the expense of a charitable society, even if you are financially capable, as long as the society does not stipulate financial inability as a condition for performing Umrah at its expense. You are permitted to perform Umrah on behalf of another person after you have performed it for yourself, and you are hoped to receive the reward of Umrah.