Is the aunt's statement, "If so-and-so proposes to her daughter, I will buy the wedding ring," considered a vow, and is she sinful if she does not buy it?

The statement made by your aunt is not considered a valid vow (nadhr shar'i) because the phrasing of a vow entails a commitment, such as "It is a duty upon me for the sake of Allah to do such-and-such." Consequently, your aunt is not obligated to buy the ring, and there is no sin upon her if she does not do so. What your aunt said falls under the category of a promise, and fulfilling a promise is recommended (mustahabb) but not obligatory (wajib) according to the majority of jurists.