Is the Udhiyah binding on the sister who did not purchase it, and her husband is absent?

The Udhiyah (definition) (sacrifice) is a Sunnah (definition), not obligatory, according to the majority of scholars, and your sister is not required to perform it. If she purchases a sacrifice from her own money and slaughters it on behalf of herself and her family, that would be good, and she would attain a great reward for it. However, she cannot do so using her husband's money unless he explicitly authorizes her to do so.