Question: [Sunday, 24th Nisan, 5782]
When is Yizkor to be said in years that the second day of Shavuos falls on Shabbos? On Shabbos? On Friday, the first day of Chag as said in Israel? Or not at all?
On Shabbos. Yizkor is customarily recited [in the diaspora] on the second day of Shavuos even when it falls on Shabbos.
The widespread Ashkenazi custom is to recite Yizkor on Yom Kippur and on the last day of Yom Tov [with exception to Simchas Torah in the Diaspora, as brought above]. Now, the question is raised as what one is to do when the last day of Yom Tov falls on Shabbos, and as to whether there is any prohibition to recite Yizkor on Shabbos. So, contrary to popular belief, not only is there no prohibition to recite Yizkor on Shabbos, but furthermore, the custom used to be to recite Yizkor on every Shabbos for all souls within 12 months from death, or by the Shabbos prior to the Yahrzeit. The reason it is allowed on Shabbos is because Yizkor is not considered a supplication, which is forbidden on Shabbos, but rather a blessing, and is thus allowed. Now, although we are no longer accustomed in doing so today, nonetheless, no prohibition exists, and hence we recite Yizkor even when the last day of Yom Tov falls on Shabbos. Nonetheless, in previous times there were many communities who extended the custom of no longer saying Yizkor on Shabbos even to when the last day of Yom Tov fell on Shabbos. Now, while this custom of not saying Yizkor when Yom Tov falls on Shabbos is recorded in previous Poskim as being a custom of some communities, practically the more recent Poskim of today conclude that the widespread custom is to recite it, as is allowed according to the letter of the law. Nonetheless, obviously each community is to follow their custom, and hence in the above Shul that you were in, seemingly they still follow the old custom of some communities of not saying it on Shabbos.
Sources: See Encyclopedia Talmudit Vol. 8 Erech Hazkaras Neshamos p. 607; Poskim who write to say it on last day of Yom Tov even when falls on Shabbos: Implication of Kitzur SHU”A 133:21 who does not differentiate in this matter; Custom of some brought in Shaareiy Ephraim 10:31 and in Pischeiy Shearim ibid 33 that so was the custom of the Arugas Habosem and so is the custom in his own Beis Midrash; Mateh Efraim 621:2; Lashon Chachamim 28; Darkei Chaim Veshalom 804; Luach Eretz Yisrael; Darkei Chesed p. 137; Otzer Minhagei Chabad Leil Shavuos 47; Luach Devar Yom Beyomo of Belz; Halichos Chaim [Halbershtam] Shevi’I Shel Pedsach 7; Nitei Gavriel 78:9 that so is the custom in our communities; Piskeiy Teshuvos 284:14; Poskim who write not to recite it when Yom Tov falls on Shabbos: Shaareiy Ephraim 10:31 that so is the custom; Custom of some, brought in Nitei Gavriel ibid; Teshuvah of Hisorerus Teshuvah; See regarding the custom of previous times to say Yizkor on every Shabbos: Admur 284:14; Rama 284:7; Shivlei Haleket 81; See Piskeiy Teshuvos 284:14; Nitei Gavriel 78:1; 79:18-20; See regarding the custom of no longer saying Yizkor on every Shabbos: Yizkor on Shabbos was Omitted from Admur in Siddur; Nimukei Orach Chaim 284:2; Maharam Brisk 2:52; Betzel Hachochmah 4:121; Ketzos Hashulchan ibid regarding first year, although he does record the custom before the Yahrzeit; Igros Moshe 2:74, however, does record this custom; See Piskeiy Teshuvos ibid and Nitei Gavriel 78:3 [See there also regarding Hashkava, Keil Malei Rachamim]