Question: How many people need to have finished Davening Shemone Esrei before the Chazzan can continue?
Answer: The Shulchan Aruch (OC 124:4) writes that the Chazzan should wait for 9 others to respond before beginning Chazaras Hashatz. Failing that, it is close to making a Bracha l’vatala. In OC 55:6 the Shulchan Aruch seems to contradict himself, however, by saying that if one of the 10 can’t respond as they have begun Davening or fallen asleep, they may still be included. The Achronim explain this difficulty in different ways. The Perisha and Magen Avraham (55:8) conclude that one doesn't need 9 to answer for Chazaras Hashatz (See Taz, Pri Chadash and Aruch Hashulchan 55:13), while the Shulchan Aruch Harav (55:7), Kitzur Shulchan Aruch (20:2) and Kaf Hachaim (55:48) maintain that there does need to be a Minyan.
R’ Ephraim Greenblatt (Rivevos Ephraim 3:145) explains that one only needs a Minyan when repeating Chazaras Hashatz. To say Kaddish during Maariv, however, 6 people will suffice, and one can rely on others still Davening or sleeping.
The Noda Biyehuda (Kama: EH 54) was asked about someone who was Mesader Kiddushin without checking that there was a Minyan present for the Brachos. He replied that while certainly lechatchila one needs to have a Miyan present for all Devarim Shebikedusha, Bedieved if one doesn't have a Minyan, one is still yotzei. He brings the Gemara Yerushalmi as a proof: If people leave the Davening in the middle, leaving less than a Minyan, the Chazzan may continue.
R’ Shmuel Wosner, (Shevet Halevi 4:14) however, questions this proof, differentiating between starting Chazaras Hashatz when you need a Minyan to bring the Shechina, and already having a Minyan, when the Shechina won’t leave just because one or two people do.In conclusion, one should ideally wait for 9 others to respond for Chazaras Hashatz, though can rely on the more lenient authorities when absolutely necessary (Tzitz Eliezer 12:9 and Yalkut Yosef 124:14).