There has recently been in several places in social media some discussion of baptism of vicarious desire (baptismus in voto parentum); unfortunately it has been plagued by several serious misconceptions. I think it is worthwhile to make a few clarifications, although they do not on their own rise to a full treatment.
(1) The question of whether there is baptism of vicarious desire arises from the case of infants whose parents were intending to baptize them who die before baptism. There are a few things potentially confusing about the name.
Despite the English name, the 'desire' has nothing to do with 'desire' in the colloquial sense; the 'desire' in this context specifically indicates baptismal intention. For instance, catechumens may die before they receive sacramental baptism; but to be a sincere catechumen is to be preparing for baptism, and thus to have baptismal intention, so they are said to receive baptism of desire. Baptismal intention is a necessary but not sufficient element of sacramental baptism; thus in the case of catechumens who die, they have part but not all of what makes for sacramental baptism. When this has actually been discussed, saints and doctors have typically seen this as a case of genuine participation in something not completely possessed; that is to say, baptism of desire is genuine baptism, and thus suffices for salvation, but it is incomplete and does not provide everything that sacramental baptism does (e.g., it does not give any kind of sacramental character because it is not sacramental baptism). This is known as 'baptism of desire'. It is distinct from (although related to) what we are currently considering. Despite occasional skeptics, here is no real doubt in Catholic theology that there are cases of baptism of desire; it's not a hugely common topic, but one can find clear support for it in some Church Fathers, some particular version of it seems to be the situation for the Old Testament saints, and the Council of Trent at least implies it as a possibility and is often interpreted as requiring it as part of Catholic theology of baptism.
Most forms of baptism, including baptism of desire, involve what is known as proper baptismal intention; that is to say, the person baptized is the one who has the baptismal intention itself. However, there are kinds of baptism that do not involve proper baptismal intention but vicarious baptismal intention. This is the sort of intention that is involved in sacramental baptism of infants; the baptismal intention is that of the parents and the Church on behalf of the infants rather than of the infants on behalf of themselves. There is no real doubt in Catholic theology that vicarious baptismal intention suffices in the case of sacramental baptism; infants receiving sacramental baptism with only vicarious baptismal intention (which is the only way they can) are genuinely baptized.
Thus the question of whether there is baptism of vicarious desire amounts to this: Are there cases of baptism falling short of sacramental baptism where the baptismal intention is vicarious? For example, if parents are preparing for an infant to have sacramental baptism but the infant dies before it can actually receive sacramental baptism, is the infant baptized? Catholic theology does not take baptism to be an all or nothing affair; there are baptisms that are taken to be genuinely but only incompletely or partially baptismal, like baptism of desire and baptism of blood (martyrdom). Is an infant who dies (for instance) just before receiving sacramental baptism baptized in this kind of genuine-but-incomplete sense? That is the question.
It is important to recognize, because I find that people regularly fail to do so, that the question is not whether infants can be saved without baptism; the question is whether they can be baptized without sacramental baptism. Likewise, we are not considering whether every infant is so baptized; we are considering whether infants who were going to receive sacramental baptism but did not are baptized. (There's a weird notion that occasionally floats around that baptism of vicarious desire is an alternative to limbo; in fact, they don't really have anything to do with each other -- the claim that there is a limbo of children is a claim about what happens to children who are not baptized, the claim that there is baptism of vicarious desire is a claim that some children are baptized in a particular way. Dragging limbo into the matter is an ignoratio elenchi.)
(2) Some peope take the fact that there is nonsacramental baptism due to proper baptismal intention and sacramental baptism due to vicarious baptismal intention as directly establishing that there can be nonsacramental baptism due to vicarious baptismal intention. This is probably too quick, but it is true that it makes it a reasonable question to ask. Given that we certainly have
what principled reason is there to claim that the right corner of the table should get a NO rather than a YES? Infant baptism does establish that vicarious baptismal intention can sometimes be adequate as baptismal intention; baptism of desire does establish that sometimes one can be baptized with proper baptismal intention without having received sacramental baptism in particular. So the question becomes, what principled reason is there for denying that someone can be baptized with vicarious baptismal intention without having received sacramental baptism in particular? And that turns out to be quite difficult; most of the arguments I've come across would, if they worked, also establish that there is no infant sacramental baptism. What is given with proper intention in baptism of desire and adult sacramental baptism is given with vicarious intention in infant sacramental baptism. Since infant sacramental baptism is a non-negotiable YES in Catholic theology, any argument against baptism of vicarious desire that would also imply that infants cannot be sacramentally baptized, if one attempted to apply the same argument to sacramentally baptized infants, is a non-starter. In practice I find that critics of the idea of baptism of vicarious desire tend to start with the baptism of desire and then argue that infants don't have proper intention and so don't have baptism of desire. This is trivially true, and irrelevant, because baptism of vicarious desire is not baptism of desire in this way. If lack of proper baptismal intention were sufficient, no infants could receive any kind of baptism. The fundamental puzzle that has to be addressed if one rejects baptism of vicarious desire is how the arguments for infant sacramental baptism work if vicarious baptismal intention is not adequate for baptism. (An indirect version of this, which one finds very occasionally discussed in Baroque authors, is circumcised children in the Old Testament, who are taken to have baptism by anticipation, a very specific form of baptism of desire, but who, if they died before the age of reason couldn't be baptized, as adult Old Testament saints could be, under baptism of desire, since their anticipation of Christ was vicarious rather than proper.)