Is holding hands and forming a circle around the altar during eucharistic prayer allowed?

Is it permitted to form a circle around the altar and hold hands during the Eucharistic Prayer?

The short answer is: No.

When I was in the major seminary, we would gather around the altar once a week (Saturdays) during the Eucharistic Prayer, but we did not hold hands. Apparently Canon Law is silent on this subject, which is sometimes considered a local custom. The seminary, later abandoned this custom, when a rather traditional student refused to join in and remained in the pews. The rector tried to encourage the seminarian to make some gesture of fellowship with the others, but to no avail.

The student in question thought the custom mimicked the actions of the priests at Mass and stayed firm in his resolve not to join in.

Today, he is a priest in the Priestly Fraternity of Saint Peter and is highly respected by his former seminary professors, even though the seminary at that time was not a traditional one.

If the priest is holding hands with the Children during the Eucharistic Prayer than that might pose more serious problems, because the priest is obliged to perform certain gestures like laying his hands over the gifts to be consecrated, to make the sign of the cross and to elevate the Body and Blood of Christ at the appropriate times. These gestures are non-negotiable and must be performed uniquely by the priest only.

It may not be in Canon Law, but Rome has spoken against it:

Although there is no precedent in Catholic liturgical tradition the question of whether it is permissible for children, youth or adults to stand around the altar during the Eucharistic Prayer continues to be asked. The argument often given by those who encourage this practice is to foster "community". In 1981 the Congregation for Divine Worship addressed this question in its official journal Notitiae. In an official interpretation of no. 101 in the General Instruction of the Roman Missal (GIRM) it responded as follows:

Query: At the presentation of 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 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]

The necessity of preserving the proper distinction of roles was again addressed in 1997, as part of a larger problem of growing confusion between the roles of the ordained and the non-ordained. In that year, the Roman Congregations of Clergy, Doctrine of the Faith, Bishops, Divine Worship, Religious, Laity and Evangelization, as well as the Pontifical Council for the Interpretation of Legislative Texts, jointly issued a document intended to recall the Church to the practice of a clear distinction between the role of the laity and that of the clergy. Called the Instruction On Certain Questions Regarding The Collaboration Of The Non-Ordained Faithful In The Sacred Ministry Of Priest it noted the confusing practices present in the Church today which do not respect the theological distinction between those in Holy Orders and those who are not. On the one hand the activity of the laity is a sign of the vitality of the Church in attempting to live the teaching of Vatican II, which calls us to an active role in the liturgy and a greater role in the mission of the Church. On the other, it is a sign that some have forgotten necessary and basic distinctions that reflect the different sacramental meanings of different vocations in the Church. - Standing Around the Altar

Her are a few more sources on holding hands at Mass and the Our Father in particular:

Holding Hands at Mass During the “Our Father”

Should people hold hands during the Our Father at Mass?