Is it permissible for the family of a murdered person to seek revenge on their relative's killer, who was not executed due to the suspension of Sharia rulings? And is a person killed during a robbery considered a martyr whose sins are forgiven?

The family of the slain is not permitted to carry out retribution themselves without recourse to the ruling authority.

As for whoever is killed in defense of their property, life, religion, family, or against injustice, they are a martyr. This type of martyr is a martyr of the Hereafter; they are not treated in this world like martyrs [of battle], so they are washed, shrouded, and prayed over. However, they attain the reward of martyrs in the Hereafter.