Gemara

(Beraisa): If Reuven was pursing Shimon to kill him, we tell him 'you are chasing a Yisrael. He is a Ben Bris...'

Bava Metzia 71b - Version #1 - (Rav Ashi): A Nochri cannot be (or make) a Shali'ach only for Terumah, but for other matters, he can!

Rejection: That is wrong! A Nochri cannot be a Shali'ach for Terumah, for we learn from "Gam Atem" - just like you are Bnei Bris (bound by the covenant of circumcision, i.e. Yisrael or a slave), also your Sheluchim must be. We learn from Terumah that Sheluchim work in general!

Version #2 (Rav Ashi): "Gam Atem" teaches that just like you are Bnei Bris, also your Sheluchim must be. This is our source that Sheluchim work. We learn that only Atem (Yisrael) can have Sheluchim!

Avodah Zarah 27a (Daro bar Papa): "V'Atah Es Brisi Tishmor" teaches that a Nochri cannot circumcise.

(R. Yochanan): We learn from "Himol Yimol" (one who is circumcised may circumcise).

Question: What is the difference between these opinions?

Answer #1: They argue about a Yisrael whose brothers died through circumcision (he is exempt from circumcision due to the danger). He is commanded, but he is Arel (uncircumcised).

Rejection: Even an Arel Yisrael is considered circumcised!

Answer #2: Rather, they argue about a Yisraelis.

Rishonim

Rambam (Hilchos Shluchim 2:1): A Nochri cannot be a Shali'ach or make a Shali'ach for anything in Torah. It says "you will take Terumah Gam Atem" - just like you are Bnei Bris, also your Sheluchim must be.

Rambam (Hilchos Gerushin 3:15): Anyone may write a Get except for for five: a Nochri, slave, Cheresh, lunatic or minor. Even a woman can write her own Get. A Yisrael who is Meshumad (serves idolatry) or is Mechalel Shabbos in public is like a Nochri in every way.

Rambam (16): A Nochri is Pasul because he writes for his own sake, and a Get must be Lishmah. A slave is Pasul because he is not Ben Krisus (divorce does not apply to him).

Ri'az (in Shiltei ha'Giborim Gitin 12a): The Rambam says that a Mumar or a Mechalel Shabbos is like a Nochri in every way. I say that just like his Kidushin or Get takes effect, he can be a Kosher Yisrael's Shali'ach to divorce.

Tosfos (Sanhedrin 72b DH Yisrael): 'Ben Bris' connotes that he is not a Mumar to serve idolatry. We seek to kill a Mumar, and do not save him!

Poskim

Shulchan Aruch (EH 123:2): (Regarding writing a Get) a Mumar or one who is Mechalel Shabbos in public is like a Nochri in every way.

Beis Shmuel (5): The Rambam disqualifies a Nochri because he writes for himself, and a slave because he is not Ben Krisus. He did not disqualify a Nochri because he is not Ben Krisus, for a Mumar is included in Nochri, and a Mumar is Ben Krisus.

Shulchan Aruch (EH 140:11): if a Mumar sends a Get to his wfe, we tell him that when he gives it to the Shali'ach, his wife is divorced, and for a clearer Heter, he should tell him 'be my Shali'ach to give it to my wife.'

Shevus Yakov (1:114): The Shulchan Aruch here and in Siman 141 says 'some say that', but I did not find anyone who disagrees.

Shach (CM 126:92): The Ri says that Ma'amad Sheloshtam does not work with a Nochri. Maharshdam (CM 359) says that similarly, it does not work with a Mumar. This is wrong. The Gemara and Poskim hold that a Mumar has Zechiyah, e.g. EH 140:5!

Note: EH 140:5 bring two opinions about whether one may be Zocheh a Get from a Mumar for his wife.

Shulchan Aruch (141:33): Someone Pasul for testimony mid'Oraisa may not even be a Shali'ach to bring a Get unless the signatures were validated. Some disqualify even in this case.

Rema: A Mumar is like one Pasul mid'Oraisa.

Magen Avraham (OC 189:1): A Mumar is not considered Ben Bris (Tosfos Sanhedrin 72b DH Yisrael). The Rema disqualifies a Mumar like a Pasul witness. This implies that he can be a Shali'ach. According to Tosfos, he cannot! Perhaps Tosfos discusses a Mumar who wants to anger Hash-m, and the Rema discusses one who sins for pleasure.

Beis Shmuel (47): The Rema is like Ri'az. Since a Mumar is a Ben Bris and Gitin and Kidushin apply to him, he can be a Shali'ach. He cannot write a Get because he does not write Lishmah. Tosfos (Sanhedrin 72b) and the Rambam do not consider him to be a Ben Bris.

Chachmas Shlomo: We learn from Avodah Zarah 27a, which suggested (and rejected) that the opinions argue about a Yisrael whose brothers died through circumcision. Why didn't it say that they argue about a Mumar who refuses to circumcise? We must say that "v'Atah Es Brisi Tishmor" does not apply to a Mumar to circumcision. This shows that he is not a Ben Bris, and all the more so a Mumar to the entire Torah. Another proof is from Bechoros 30b, which says that a convert suspected about one matter is suspected about everything, and is like a Yisrael Mumar. This teaches that if he was Mekadesh a woman, she is Mekudeshes. It did not say that this teaches about Shlichus! We infer that even if he is not considered a Nochri, a Mumar cannot be a Shali'ach. However, a Mishnah (Eruvin 31b) says that one who does not admit to an Eruv cannot be a Shali'ach for an Eruv. Rav Chisda says that it refers to a Kusi. Why didn't he say a Tzeduki or a Mumar to be Mechalel Shabbos? Rav Chisda holds that the problem is that he lacks power to acquire Shevisah (residence for Shabbos); a Mumar can acquire Shevisah. We rule unlike him; we hold the problem is that he is not trusted. In any case, we learn from here that a Mumar can be a Shali'ach!

Rema (141:59): Some say that a Mumar cannot make a Shali'ach to give a Get, for we are concerned lest he be Mevatel it. Some say that we do like it says in 140:11.

Beis Yosef (Teshuvah, Gitin 14, citing Kaftor va'Ferach 10 Dinei Meshumad): One should not allow a Mumar to make a Shali'ach to bring a Get when there is an alternative. Riva says that he cannot make a Shali'ach. It seems that he is not Ben Bris, for Torah and Mitzvos are called Bris. Only a Mumar who transgresses for pleasure is called 'your brother'. Even according to the opinion that one who transgressed to anger Hash-m is only a Mumar (but is not like a Nochri), one who serves idolatry is like a Nochri. The Sefer (Yevamos 16b) says that if a Nochri was Mekadesh nowadays, we require a Get. He must give it directly to her or her Shali'ach, for he cannot make a Shali'ach. Also R. Yechi'el and R. Eliezer of Garmiza say so.

Beis Yosef (ibid., citing Menachem ha'Bavli): It seems that the Rambam agrees. He did not say 'a Meshumad or one who is Mechalel Shabbos in public cannot write a Get', rather, 'he is like a Nochri.' This implies that he is like a Nochri in every way regarding Gitin, e.g. writing and Shlichus. Do not say that this is only for others, but he can write and send his own Get, since Gitin applies to him. He wrote 'a woman can write her own Get' just before 'a Meshumad or one who is Mechalel Shabbos in public is like a Nochri in every way', i.e. even regarding his own Get. Also, in Hilchos Shluchim the Rambam says that only a Ben Bris can be a Shali'ach. Rashi says that a Ben Bris is circumcized, and is Ben Bris b'Mitzvos. The Halachah always follows the second version. Therefore, the Rambam learns that also a Meshale'ach must be a Ben Bris. Some allow a Mumar to make a Shali'ach. They hold that Ben Bris is one who already accepted the yoke of Mitzvos, even if he does not observe them now. We learn from "v'Shilach" that a man can make a Shali'ach to divorce his wife.

Beis Yosef (ibid.): I say that we cannot learn from Shlichus in general to divorce, for Gitin applies to a Mumar. The proof from a Nochri who was Mekadesh is invalid, for the Halachah does not follow this opinion. Also, we are concerned for his Kidushin because perhaps he is a Yisrael. If he is, he has Shlichus (unless we say that a Mumar has no Shlichus, which is the question at hand)! R. Yechi'el does not allow Shlichus, lest he be Mevatel it. R. Eliezer of Garmiza forbids a Mumar to bring another's Get, lest people think that he can send his own Get through a Shali'ach, but we forbid this, lest he be Mevatel it. The Rambam in Hilchos Gerushin discusses only writing a Get. We cannot learn to Shlichus to give a Get. The proof from Hilchos Shlichus is invalid. Perhaps divorce is different, for it applies to him. I say that all agree that he is not called Ben Bris, but he could be a Shali'ach for divorce since it applies to him, if not for concern lest he be Mevatel.

Eshel Avraham (OC 448:4): L'Chatchilah, one may not make a Mumar a Shali'ach to sell one's Chametz. B'Di'eved, it is surely permitted.