Can Orthodox Church members be extraordinary ministers in the Roman Catholic rite? It is known that members of the Eastern Orthodox Church who are not formally joined to the Roman Catholic Church can receive Holy Communion and some other mysteries (such as Confession) without any obstacles.

Can the same members of the Orthodox Church who consciously receive Holy Communion and do confession in the Roman Catholic Church but are not formally joined to the Catholic Church take part in extraordinary ministry? If not, which Canons restrict them from doing so, and why?
I can find no specific rule forbidding it, but almost certainly not.
The document Immensae Caritatis, written in 1973 by what was then the Sacred Congregation of the Sacraments first raised the possibility of lay extraordinary ministers of the Eucharist. The precise text used is:

In order, then, that the faithful who are in the state of grace and rightly and devoutly wish to share in the sacred meal may not be deprived of this sacramental aid and solace, Pope Paul VI has decided it opportune to authorize special ministers who will be empowered to give communion to themselves and others of the faithful, under the exact and specified conditions here listed.
I. Local Ordinaries possess the faculty enabling them to permit fit persons, each chosen by name as a special minister, in a given instance or for a set period or even permanently, to give communion to themselves and others of the faithful and to carry it to the sick residing at home:
a. whenever no priest, deacon, or acolyte is available;
b. whenever the same ministers are impeded from administering communion because of another pastoral ministry, ill-health, or old age;
c. whenever the number of faithful wishing to receive communion is so great that the celebration of Mass or the giving of communion outside Mass would take too long.
... The faithful who are special ministers of communion must be persons whose good qualities of Christian life, faith, and morals recommend them. Let them strive to be worthy of this great office, foster their own devotion to the Eucharist, and show an example to the rest of the faithful by their own devotion and reverence toward the most august sacrament of the altar. No one is to be chosen whose appointment the faithful might find disquieting.

This last sentence already raises a concern. If any of those receiving Communion is aware that the minister is not a Catholic, what will he or she think about the importance of being Catholic in the Eucharist? Might they conclude that anyone, regardless of their religion, could administer the Sacrament?
Ultimately, as this document says, it's the local ordinary (usually the bishop of the diocese) who has the final say. The bishops already have a say over who receives Communion—the US Conference of Catholic Bishops, for example, wrote a document in 1996 titled &quot;Guidelines for the Reception of Communion&quot;. It states that

Members of the Orthodox Churches, the Assyrian Church of the East, and the Polish National Catholic Church are urged to respect the discipline of their own Churches.

The &quot;discipline&quot; referred to, that is, the Orthodox judgment of whether an Orthodox Christian may receive the Eucharist at a Catholic Mass, appears to be that such may not happen:

Orthodox Christians are not permitted to receive Communion in non-Orthodox communities, including the Roman Catholic. To do so would imply a unity that in fact does not yet exist.

(&quot;Communion in Roman Catholic Church&quot;, website of the Orthodox Church in America)
It would seem odd indeed if the Catholic Church would respect Orthodox wishes not to receive the Eucharist, and yet appoint a member of the Orthodox Church to distribute it.
Ultimately, the question is up to the bishop, who enjoys a great deal of latitude in these sorts of issues, but I'd be very surprised indeed if such a thing ever happened—if such a minister was ever appointed, and more so if the minister accepted.