If a batter is injured, while hitting a home run out of a park, and cannot physically run the bases, can a pinch runner be used?

If a batter-runner is incapacitated during a home run (or any other play where he is entitled to a particular base, such as a walk, a ground rule double, or a ball thrown into the dugout), then his team may use a substitute runner (who then takes his place in the batting order and on the field, unless further replaced). See the MLB Official Baseball rules, from rule 5.12:

When an accident incapacitates a player or an umpire;

(A) If an accident to a runner is such as to prevent him from proceeding to a base to which he is entitled, as on a home run hit out of the playing field, or an award of one or more bases, a substitute runner shall be permitted to complete the play.