Is it permissible for someone performing Hajj at-Tamattu‘ to perform ‘Umrah on behalf of another person, or a voluntary ‘Umrah for himself, during the period between the ‘Umrah and the Hajj?

It is permissible to repeat Umrah during a single journey, but it is not from the Sunnah. The Sunnah is to perform only one ritual, following the example of the Prophet (peace and blessings be upon him). It is better for someone who wishes to perform Umrah on behalf of another person to supplicate for them.