Is it permissible to eat what is hunted by a relative who does not pray, knowing that he may not mention the name of Allah before hunting?

One who abandons prayer is a disbeliever. You must strive to advise him and remind him of the seriousness of his action. If he persists after the evidence has been established against him, then shun him. For game to be lawful, the hunter must be someone whose slaughtered animal is permissible to consume, such as a Muslim or a person of the Book. Therefore, it is not permissible for you to eat what your relative, who abandons prayer, hunts, because he is an apostate disbeliever, and thus his slaughtered animal and his game are not permissible.