When you die without a Will, the law including the Wills, Estates and Succession Act governs the manner in which your assets are distributed. While you can apply to the court to deviate from these guidelines, courts may be reluctant to do so without written wishes of the Deceased.
Anyone in your family can apply to be the executor (known as Estate Administrator when there is no Will), and can likely do so, subject to some exceptions for example: a family member disputing the application.