Is it permissible for the mother to sell the farm, and is the sale in this manner religiously sanctioned or void, and do the children have a legitimate right to the farm and the vacant land whose ownership was transferred to the eldest brother, knowing that the deceased sister has not yet received her due?

First, the inheritance is a right for all heirs, and it is not permissible to dispose of anyone's share except with their consent. The orphan's money must be preserved and managed in the most beneficial way, and their guardian is their grandfather, their trustee, or the judge. If an heir dies before the division, their heir takes their place.

Regarding the farm, if it was built with the effort of the eldest brother and the mother's money, and it included a portion of the money from the agricultural land owned by the heirs, then if this portion was only from the share of the eldest brother or the mother, then none of the children have a right to it. However, if it contained anything from the share of one of the heirs without their permission, then the eldest brother must return the money to its owner, and his taking it at his mother's command does not change the ruling. If it is proven that the farm belonged to the mother and she authorized her eldest son to sell it for Hajj with its price, and he sold it for its equivalent price and then bought it, there is no harm, and it becomes his property. The mother's gift of the remaining price of the farm to her eldest son's children is permissible.

As for the vacant land, it is not part of the inheritance if it was not owned by your father at the time of his death. Rather, it belongs to whoever bought it from the government, unless the government bought it for the eldest brother as a representative of his brothers with the heirs' money, in which case they have a right to it.

The matter of inheritances is serious and complex, and a fatwa alone is not sufficient. Rather, it is necessary to consult scholars in person or refer the matter to Sharia courts.