Is it permissible to add "and on behalf of the Muslims" when naming the sacrifice as being on my behalf and on behalf of my family, and will the one who does so receive their reward?

The dedication of the reward of the sacrifice (Udhiyah) to others is permissible, and the reward will reach them. It is also permissible to include both the living and the deceased in the reward of the Udhiyah. However, including Muslims with you with the intention of having it suffice for them is not valid, as the sacrifice (Udhiyah) is sufficient only for the one performing the sacrifice and their household.