Is he who prevents his wife and children from visiting her family considered a severer of kinship ties and deserving of the punishments for those who sever kinship ties in the Hereafter, and to what extent is it permissible for the wife to seek a divorce for this reason?

What you intend by depriving your wife's family of seeing your children and preventing them from visiting them is an injustice and a severance of kin ties (قطع للرحم), and it is not permissible for you to prevent your wife's parents from visiting her unless you fear harm from their visit. There is no justification for divorcing your righteous wife merely because of your previous disagreement with her family; in fact, that might be an injustice to her. We advise you to hold on to your wife and treat her kindly, and to maintain her ties and your children's ties with her family, and to forgive them for any offense they may have caused you.