One way to become an Extraordinary Minister of Holy Communion is to become an instituted acolyte. These are permanent Extraordinary Ministers of Holy Communion.
The requirements for this are given in the Motu Proprio of 1972 Ministeria Quaedam:
"7. In accordance with the ancient tradition of the Church, institution to the ministries of reader and acolyte is reserved to men.
"8. The following are requirements for admission to the ministries:
“a.the presentation of a petition that has been freely made out and signed by the aspirant to the Ordinary (the bishop and, in clerical institutes, the major superior) who has the right to accept the petition;
b.a suitable age and special qualities to be determined by the conference of bishops;
c.a firm will to give faithful service to God and the Christian people.”
I think the procedure to become an Extraordinary Minister of Holy Communion were slightly changed with the Instruction of 2004, Redemptionis Sacramentum.
In 1973 the Instruction Immensae caritatis described the appointment of these “in a given instance or for a set period or even permanently”. (Documents on the Liturgy, Liturgical Press, 1982, ISBN 0-8146-1281-4, page 651, n. 2075). The 2004 Instruction changes this to “one occasion or for a specified time”. (Redemptionis Sacramentum, n. 155)
A Rite of Commissioning was also published, with the 1973 instruction: “When the local Ordinary or his delegate deputes anyone in particular circumstances as a special minister of holy communion, that person should receive a mandate according to the following rite. This rite may take place either within or outside Mass, with a congregation present.” (Documents on the Liturgy, n. 2949).
This rite was “advisable” rather than essential: “If there is time, it is advisable that the fit person chosen … should receive a commission (mandatum) according to the rite annexed to this Instruction.” (Documents on the Liturgy, n. 2080).
The 2004 Instruction highlights this, in n. 155: “This act of appointment, however, does not necessarily take a liturgical form, nor, if it does take a liturgical form, should it resemble sacred Ordination in any way.”
The appointments were made in the 1973 Instruction by the “local Ordinary or his delegate” but in the 2004 Instruction “by the diocesan bishop”.
So now to be properly appointed as an Extraordinary Minister of Holy Communion (other than an instituted acolyte) I think you need a document signed by the diocesan bishop, including an expiry date for the appointment.
However, as the 2004 Instruction has in 155: “Finally, in special cases of an unforeseen nature, permission can be given for a single occasion by the Priest who presides at the celebration of the Eucharist.”