Is Hajj considered invalid and unrewarded if performed with money earned by a son from transporting alcohol, especially if this is the sole source of livelihood? And what is the proof for that?

If your son earns only from selling alcohol, his money is unlawful (haram), and it is not permissible to accept his gift except out of necessity and to the extent of that necessity. However, if he earns from selling alcohol and other sources, his money is mixed, and it is permissible to accept his gift, though it is disliked (makruh). Your Hajj is valid in all cases, and you must advise your son to abandon this forbidden work and seek lawful employment. If he does not respond, then what you are compelled to take from his money for your essential needs is not forbidden for you, but anything beyond that is not permissible for you to benefit from.