I believe the minister in the byzantine theology is the priest.
I’ve read that, while a deacon can witness a marriage, he can not perform the crowning.
(all cited canons from the CCEO, english translation at
intratext.com/IXT/ENG1199/ and all are paraphrases.)
Cannon 796 gives the bishop permission to dispense from form or impediment in cases of death, and devolves that to the pastor when the bishop is unreachable by letter or personal contact.
Canon 828 specifies that only a priest or bishop may bless a marriage.
Canon 832 permits marriage before witnesses alone when no priest or bishop is available, but also requires that, if a non-catholic but valid priest is available and permitted by his ordinary, said non-catholic priest bless the marriage. If no blessing was given, when it becomes possible, it’s required to receive the blessing.
Canon 841 covers registering the marriage in the parish baptismal register, and notification to notate it in the baptismal register of their place of baptism
So, in cases where priests of one’s own church are unavailable
1st option: find a priest of a different Church Sui Iuris
2nd option: Find an Orthodox (EO, OO) or Assyrian Church, or PNCC priest and see if they will do so
3rd option: get the bishop’s permission for witnesses only marriage; if a deacon is available, he will almost invariably tasked by the bishop to stand witness for the Church.
Now, I’ve seen photos of a priest, from a Soviet jail window, blessing a marriage of a couple below, while a deacon held the crowns… but I’m certain that was an economia due to pogrom…