Is it permissible to give an ongoing charity (ṣadaqah jāriyah) on behalf of my sister and her children without their knowledge, with me contributing a small amount to the charity? And will the one who gives charity be rewarded if the charity is designated for them only?

It is permissible to give a perpetual charity (Sadaqah Jariyah) on behalf of your sister and her household, even without their knowledge. The one who gives the charity and those on whose behalf it is given will both be rewarded if the intention is for the charity to be for them.

Ibn al-Qayyim said: "Whoever fasts, prays, or gives charity, and dedicates its reward to another, whether deceased or living, it is permissible. Its reward reaches them according to the Ahlus-Sunnah wal-Jama'ah. The one who performs the act also receives reward through his intention for them before the act, whether he explicitly dedicates it or not. However, it is better for the one performing the act of obedience to dedicate it to himself."

Both the one who dedicates the reward and the one to whom it is dedicated will be rewarded.