Is it permissible to give zakat to a poor grandmother who has no permanent provider, knowing that her salary from the Palestinian Authority for mothers of martyrs is insufficient for her needs, and is the zakat considered Islamically acceptable in this case?

It is permissible to give zakat to a poor grandmother according to the Malikis, while it is not permissible according to the Hanafis and Shafi'is if she is not self-sufficient without your maintenance, nor according to the Hanbalis due to the obligation of her maintenance upon you. If you are not qualified to weigh the opinions, it is permissible for you to adopt the view of those who permit it; however, it is more cautious not to give her zakat, but rather to complete her needs from your maintenance.