On the matter of the form of the wedding: here’s what canon law requires:
THE FORM OF THE CELEBRATION OF MARRIAGE
Can. 1108 §1. Only those marriages are valid which are contracted before the local ordinary, pastor, or a priest or deacon delegated by either of them, who assist, and before two witnesses according to the rules expressed in the following canons and without prejudice to the exceptions mentioned in cann. ⇒ 144, ⇒ 1112, §1, ⇒ 1116, and ⇒ 1127, §§1-2.
§2. The person who assists at a marriage is understood to be only that person who is present, asks for the manifestation of the consent of the contracting parties, and receives it in the name of the Church.
Can. 1109 Unless the local ordinary and pastor have been excommunicated, interdicted, or suspended from office or declared such through a sentence or decree, by virtue of their office and within the confines of their territory they assist validly at the marriages not only of their subjects but also of those who are not their subjects provided that one of them is of the Latin rite.
[snip of details on the minister’s status]
Can. 1115 Marriages are to be celebrated in a parish where either of the contracting parties has a domicile, quasidomicile, or month long residence or, if it concerns transients, in the parish where they actually reside. With the permission of the proper ordinary or proper pastor, marriages can be celebrated elsewhere.
Can. 1116 §1. If a person competent to assist according to the norm of law cannot be present or approached without grave inconvenience, those who intend to enter into a true marriage can contract it validly and licitly before witnesses only:
1/ in danger of death;
2/ outside the danger of death provided that it is prudently foreseen that the situation will continue for a month.
§2. In either case, if some other priest or deacon who can be present is available, he must be called and be present at the celebration of the marriage together with the witnesses, without prejudice to the validity of the marriage before witnesses only.
Can. 1117 The form established above must be observed if at least one of the parties contracting marriage was baptized in the Catholic Church or received into it and has not defected from it by a formal act, without prejudice to the prescripts of ⇒ can. 1127, §2.
http://www.vatican.va/archive/ENG1104/__P40.HTM
It’s pretty darned explicit. If they the OP and their betrothed do not get married in the presence of a Catholic Cleric, they are not married validy; an invalid marriage is automatically also illicit. And sex outside a valid marriage is grave matter.
There is dispensation from 1108 available, but they are baptized and confirmed… and thus bound by Canon Law.