"My older sister does not want to stay at home."
For close personal relations you don't need 的，so 我姐姐不要留在家里 should also be correct.
Not only 的 is not required for personal relations, 不要 is not the correct translation for 'does not want'. 不想 would've been more accurate.
In other topics 不要 is used to convey the idea of 'must not', which does not seem to me to reflect the meaning of the English here.