Does sin befall a woman for refusing to care for her sister's sick children for a year, while their mother wishes to travel for work and their father lives in another country?

There is no sin upon you for refraining from caring for your sister's children. It is an individual obligation upon the mother to custody and care for her children if there is no one else, as stated in the Kuwaiti Fiqh Encyclopedia: "Custody is an obligation by Sharia; because the one in custody may perish or be harmed by neglect." Your sister should be advised to fear Allah and remain with her children, for the Prophet (peace and blessings of Allah be upon him) said: "All of you are guardians and are responsible for your subjects."