Answer:
C- Who might once have lost their ability to participate in their sports due to catastrophic injury
Step-by-step explanation:
a) that might at one time have lost their ability to participate in their sports due to catastrophic injury : ''that'' is not used to refer to a person therefore, this is wrong.
B) that due to catastrophic injury might once have lost their ability to participate in their sports : ''that'' is not used to refer to a person therefore, this is wrong.
C) who might once have lost their ability to participate in their sports due to catastrophic injury : this is correct as the right modifier "due" is placed next to the noun phrase "ability to participate in their sports''
D) who due to catastrophic injury might have at one time lost their ability to participate in their sports : the modifier "due to" is not placed next to the phrase it is modifying
E) who, due to catastrophic injury, might at one time have lost their ability to participate in their sports : he modifier "due to" is not placed next to the phrase it is modifying