What are the requirements for godparents in the Roman Catholic Church?

^{Disclaimer: Though I am a Catholic, I am not a canon lawyer, nor a theologian, nor a priest. I am reasonably certain that my answer is correct and in accordance with Church law; but if I am in error I will gladly receive correction.}

Unfortunately, it appears that such a couple as you describe is not eligible to be godparents at a Catholic baptism.

The Code of Canon Law for the Latin Church has a whole chapter on godparents—"sponsors" is the technical term—but before we go there, let's talk about the Catholic theology of baptism.

The Catholic understanding of baptism differs from that of some Protestant denominations in that Catholics consider baptism as a necessary thing. The Catechism of the Catholic Church calls baptism

the basis of the whole Christian life, the gateway to life in the Spirit ... and the door which gives access to the other sacraments. Through Baptism we are freed from sin and reborn as sons of God; we become members of Christ, are incorporated into the Church and made sharers in her mission.

(paragraph 1213)

Describing what the Sacrament of Baptism actually does, the Catechism explains:

it signifies and actually brings about the birth of water and the Spirit without which no one "can enter the kingdom of God." [cf. Titus 3:5, John 3:5]

(paragraph 1215; emphasis added)

This being the case, the Church reasons, why would one not want to baptize one's children at as early an age as possible?

Born with a fallen human nature and tainted by original sin, children also have need of the new birth in Baptism to be freed from the power of darkness and brought into the realm of the freedom of the children of God, to which all men are called. The sheer gratuitousness of the grace of salvation is particularly manifest in infant Baptism. The Church and the parents would deny a child the priceless grace of becoming a child of God were they not to confer Baptism shortly after birth.

Christian parents will recognize that this practice also accords with their role as nurturers of the life that God has entrusted to them.

(paragraphs 1250, 1251)

The Church therefore considers it very important to ensure both that children are baptized and that baptism is used as an introduction to and support for their Christian and Catholic life. To that end, they have several canons discussing baptism, who can be baptized, requirements for parents of children being baptized, and (importantly for our question) requirements for baptismal sponsors. These requirements are essentially the same in the Latin Church's Code of Canon Law as they are in the Code of Canons for the Eastern Churches, which governs the Eastern Catholic sui iuris churches. I'll review the Latin Church's rules.

The rules appear in Book IV, Part I, Title I, Chapter IV of the Code of Canon Law, "Of Sponsors".

The sponsor at an infant or child baptism has two functions: first, to stand as a witness for the parents' desire to have their child baptized; and second, to help the child grow up leading a Christian and Catholic life (cf. canon 872). With this in mind, the Church imposes restrictions on potential sponsors intended to ensure that they can appropriately fulfill this second function. The restrictions appear in canon 874, section 1. Each sponsor must:

1. be designated by the one to be baptized, by the parents or the person who takes their place, or in their absence by the pastor or minister and have the aptitude and intention of fulfilling this function;

This restriction makes sure that the sponsor is close enough to the family that they will be easily able to interact with the child, and that they are willing and able to do what a sponsor needs to do.

2. [They must] have completed the sixteenth year of age, unless the diocesan bishop has established another age, or the pastor or minister has granted an exception for a just cause;

This is to make sure that they are psychologically mature enough to ensure that the child does what a Catholic needs to do—even when it's not easy for the child, and when it's not easy for the sponsor.

3. [They must] be a Catholic who has been confirmed and has already received the most holy sacrament of the Eucharist and who leads a life of faith in keeping with the function to be taken on;

This is the restriction at the core of your question. The restriction here is to make sure that the sponsor understands what, exactly, is involved in leading a Catholic life, and in helping another to do so.

4. [They must] not be bound by any canonical penalty legitimately imposed or declared;

In other words, they have to be in good standing with the Church, and not be at risk of passing on any opinion to the child, or engendering any opinion in the child, which might not be congruent with Church doctrine.

5. [They must] not be the father or mother of the one to be baptized.

This is so that the child gets a "second chance" in the event that the parents are (or become) unable or unwilling to raise the child Catholic.

Because the couple you mention do not fulfill the requirements of part (3), they would unfortunately be ineligible to be godparents.

It may be possible for the priest to ask the bishop to dispense from that particular canon; I would imagine, given the importance the Catholic Church attaches to baptism, that a dispensation would probably not be granted even if it's possible.