When one is discussing the prayer of Shalom Aleichem one should be aware of at least two facts. On the one hand Shalom Aleichem is a relatively new prayer in Jewish terms, being only several hundred years old. On the other hand it does somewhat parallel a passage in the Talmud, tractate Shabbat
The passage as translated by Soncino states:
It was taught, R. Jose {Yosi} son of R. Judah {Yehuda} said: Two ministering angels accompany man on the eve of the Sabbath from the synagogue to his home, one a good [angel] and one an evil [one]. And when he arrives home and finds the lamp burning, the table laid and the couch [bed] covered with a spread the good angel exclaims, ‘May it be even thus on another Sabbath [too],’ and the evil angel unwillingly responds ‘amen’. But if not, the evil angel exclaims, ‘May it be even thus on another Sabbath [tool,’ and the good angel unwillingly responds, ‘amen’. {end of quote}.
The Talmudic passage however, does not mention people making requests from these 2 angels. It just gives us a spiritual window into what the angels are doing.
In light of the prohibition to pray to angels, what alternative explanation do we have for the words in Shalom Aleichem, "Bless me for peace, angels of peace"?
My gut feeling, to get out of this problem, is to contend that the words bless me for peace, angels of peace is not a request (which is forbidden), but an order or command. And indeed I found confirmation for this viewpoint in Siddur Beit Yaakov of Rabbi Yaakov Emdeen {some spell the name, Emden) in his commentary on Shalom Aleichem.
And I have at least 2 precedents that giving orders to an angel is not part of the prohibition of praying to angels. The first precedent involves the story of Rabbi Pinchas Ben Yair in the Talmud tractate Chulin, page 7a, who according to Tosafot, gave a command to an angel to split the river that was blocking his way, because he needed to get to the other side, in order to redeem captives as well as a few other important deeds.
In the Torah itself Breishit/Genesis 32:27 we find Yaakov /Jacob struggling with someone, who the sages identified as an angel. He tells the angel that he would not release him, unless he blesses him. In that situation we see he was clearly compelling the angel and not requesting a request from him.
In the case of Yaakov, in what way did compelling the angel to make a blessing help?
Rashi, says the angel wanted to wait until Yaakov would sacrifice sacrifices at Bet-El before blessing him together with the other angels.
Yaakov successfully demanded that he be blessed immediately.
Perhaps, this is the benefit, that the composer of Shalom Aleichem sought for those that recite, "Bless me for peace, angels of peace". Namely, we demand that the Angels immediately bless the Jews when coming home from synagogue Friday night, instead of taking the risk, that if they wait long enough, something might come up to stop the blessing from ever coming to be.
Alternatively, the composer of Shalom Aleichem, might have believed that saying Shalom Aleichem has no real effect on the angels decision to bless or not to bless. However, by singing the song, it draws attention to the spiritual reality of the good and bad angel mentioned in tractate Shabbat, and this by itself is worthwhile.