May members of the congregation stand around the altar during the Eucharistic prayers?

Answer: NEGATIVE
Authoritative: YES 

Not far from my home there is a student parish run by the Jesuits. The priests there are legendary for calling the congregation up to the altar during the Eucharistic Prayers; the congregation is invited to stand around the altar in a semi-circle and extend their hands during the prayers. This is said to be an expression of the reality that the Eucharist is a celebration of the entire community, not the priest alone.

This abuse is particularly egregious; most readers of this website will not attend parishes where this is an issue. Still, it is important to understand the juridical reasons why this practice is forbidden and be able to cite "chapter and verse", so to speak, to those who think it is acceptable.

The Holy See actually issued a statement on this abuse back in 1981 after receiving an inquiry concerning the practice. We repost both the query and the answer of the Holy See in their entirety:

Query: At the presentation of the gifts at a Mass with congregation, persons (lay or religious) bring to the altar the bread and wine which are to be consecrated. These gifts are received by the priest celebrant. All those participating in the Mass accompany this group procession in which the gifts are brought forward. They then stand around the altar until communion time. Is this procedure in conformity with the spirit of the law and of the Roman Missal?

Reply: Assuredly, the Eucharistic celebration is the act of the entire community, carried out by all the members of the liturgical assembly. Nevertheless, everyone must have and also must observe his or her own place and proper role: "In liturgical celebrations each one, minister or layperson, who has an office to perform, should do all of, but only, those parts which pertain to that office by the nature of the rite and the principles of the liturgy" [SC art. 29]. During the liturgy of the Eucharist, only the presiding celebrant remains at the altar. The assembly of the faithful take their place in the Church outside the presbyterium, which is reserved for the celebrant or concelebrants and altar ministers [Notitiae 17 (1981) 61].

This clearly rules out the practice outlined above, whereby the congregation stands around the altar during the Eucharistic prayers. Note also, however, that this also rules out the more common practice whereby Extraordinary Ministers of Holy Communion Eucharistic ministers are invited to stand around the altar during consecration, the statement about "altar ministers" referring to servers, not Extraordinary Ministers.

We must also remember, as the response notes from Sacrosanctum Concilium 29, that there is a proper order or hierarchy to how liturgical worship is carried out, and that attempts to deviate from this order in the name of a false egalitarianism are totally misplaced. This is well attested from the Church's tradition and pertinent magisterial documents, but we could also cite the testimony of Sacred Scripture. Consider this little-known but extremely relevant passage from the Second Book of Chronicles. We are told in 2 Chronicles 26 about the victories of the Judean King Uzziah, who won many glorious victories in war and filled Jerusalem with monuments:

"And he made devices in Jerusalem, invented by skillful men, to be on the towers and the corners, to shoot arrows and large stones. So his fame spread far and wide, for he was marvelously helped till he became strong" (2 Ch. 26:15).

But in his glory he became prideful, and this caused him to covet the special role of the priests of God in offering sacrifice at the Temple in Jerusalem:

"But when he was strong his heart was lifted up, to his destruction, for he transgressed against the Lord his God by entering the temple of the Lord to burn incense on the altar of incense. 17 So Azariah the priest went in after him, and with him were eighty priests of the Lord—valiant men. 18 And they withstood King Uzziah, and said to him, “It is not for you, Uzziah, to burn incense to the Lord, but for the priests, the sons of Aaron, who are consecrated to burn incense. Get out of the sanctuary, for you have trespassed! You shall have no honor from the Lord God.” (2 Ch. 26:16-18)

The Lord was angry with Uzziah, however. Consider the punishment inflicted upon him: 

"Then Uzziah became furious; and he had a censer in his hand to burn incense. And while he was angry with the priests, leprosy broke out on his forehead, before the priests in the house of the Lord, beside the incense altar. And Azariah the chief priest and all the priests looked at him, and there, on his forehead, he was leprous; so they thrust him out of that place. Indeed he also hurried to get out, because the Lord had struck him. King Uzziah was a leper until the day of his death. He dwelt in an isolated house, because he was a leper; for he was cut off from the house of the Lord" (ibid., v. 19-21).

There is a proper order of worship, established by God Himself. To transgress this order is a grave offense that God does not take lightly.