Does one who does not perform Hajj due to an illness that prevents him from standing and walking for a long time bear a sin, and is it permissible for him to appoint someone else to perform Hajj on his behalf?

There is no sin upon one for whom Hajj is difficult due to illness. If the illness is chronic and there is no hope of recovery, even with financial capability, then it is obligatory for him to appoint someone to perform Hajj on his behalf. However, if the illness is curable, then deputizing is not permissible. Rather, he must perform Hajj himself after recovery. The immediacy of the obligation is lifted due to his incapacitation, and he is obligated to perform Hajj himself once he recovers.