It emerges from the Gemara's discussion in Masechet Eruvin (40) that the Beracha of She'hecheyanu is recited on all the festivals, including Rosh Hashanah and Yom Kippur. (There is some controversy concerning the recitation of She'hecheyanu on the second night of Rosh Hashanah. Halacha follows the view that one must recite She'hecheyanu on the second night of Rosh Hashanah, but many people nevertheless have the practice to partake of a new fruit on this night and have in mind for the Beracha to apply to that fruit, to satisfy the view that She'hecheyanu is not required on this night.)
Generally, of course, we recite She'hecheyanu as part of Kiddush. The question thus arises, when do we recite this Beracha on Yom Kippur, when there is no Kiddush?
Our practice is for the one who leads the Kol Nidrei service on the night of Yom Kippur to recite She'hecheyanu on behalf of the congregation. Many people are unaware that the She'hecheyanu recited during Kol Nidrei pertains to the day of Yom Kippur; they erroneously assume that it refers to the new Torah scroll or the like. Both the person reciting the Beracha and the congregation must have in mind that everyone in attendance fulfills his obligation with respect to this Beracha by listening to its recitation.
If a person did not hear the recitation of She'hecheyanu during Kol Nidrei, he should recite the Beracha at any point on Yom Kippur, until sundown. The recitation of She'hecheyanu does not require the formal context of Kiddush; one may recite the Beracha wherever he is, even in the street. Therefore, a person who did not hear She'hecheyanu in the synagogue during Kol Nidrei should recite it when he remembers, wherever he is.
Summary: The Beracha of She'hechayanu must be recited on Yom Kippur just as on all other festivals. The person leading Kol Nidrei recites She'hecheyanu on behalf of the congregation, and they must all have in mind that the congregation fulfills their obligation through this recitation. One who did not hear She'hecheyanu during Kol Nidrei should recite the Beracha himself at any point until sundown on Yom Kippur.
