The rarity of non-XX/XY sexes is precisely the reason that intersex people are irrelevant in the discussion of mammalian species with more than two reproductive roles. If X0 individuals (for example) were anywhere near as common as XX or XY individuals, or played anywhere near as important a role in human reproduction as XX and XY do,
then you could usefully say that the human species has the three sexes XX, XY, and X0. Since they are neither common nor essential for reproduction, intersex people are not a third sex, but a biological accident. Just because some deviations from the XX/XY system can occasionally (one in thousands of live births) produce viable (but generally infertile) individuals does not mean mentioning those individuals brings anything useful to a discussion of a mammalian species with three equally essential reproductive functions (insemination, oviposition, incubation).
Aberrations aside, many species have only two chromosomal sexes, but multiple ‘genders’, which is another thing the OP could consider, such as the ruff, which has three types of male: territorial, satellite, and ‘faeder’. The territorial and satellite males exhibit dimorphism (from females as well as from each other), but the faeder males look similar to females. All three male genders are inherited and permanent; there is no gender-bending or hermaphroditism, nor is there any significant variation in the behavior of different males besides their courting strategy (outside the lekking season, the female look-alikes associate with other males, not with females). And re what Pedant asked, male ruffs’ gender is an autosomal polymorphism; that is, it has nothing to do with the sex chromosomes (all three male types are ZZ), and is more similar to other types of intraspecies variation like melanism or blood types. So you can have multiple genetically determined gender roles with only two allosomally determined sexes, which obviates the need to mess around with multiple sets of chromosomes, which sounds messy.
If you really want actual multiple sex-determining chromosomes, you’d be better off looking at fish than at mammals, though be aware that multiple sex chromosomes doesn’t necessarily mean you end up with multiple reproductive roles, and there will be other behavioral or morphological consequences as well. Platyfish, for example, have three sex chromosomes, but only two sex roles (females can be XX, XW, or YW, while males can be XY or YY, but all five types perform only one of the same two reproductive roles). It seems that the X chromosome also determines bright coloration, while the W chromosome does not. This means that individuals with an X are brightly colored, which is disadvantageous in females, making them vulnerable to predation, but advantageous in males, as it wins them more mates. Females with an X are therefore less likely to reproduce, while males with an X are more likely to do so. This selective pressure
for X in males ensures that the X chromosome doesn’t die out in females, and the selective pressure
against X in females ensures that the Y chromosome doesn’t die out in males, so you’re left with about equal numbers both of the ‘ideal’ XY (bright, male) and YW (dull, female), and of the ‘incidental’ XX (bright, female), XW (bright, female), and YY (dull, male).
All of that said, the OP is essentially talking about a parasitic behavior, in which neither biological parent cares for their offspring, foisting the hard work onto some other individual. The only way this can be viable such that the non-parent gender would actually be a willing partner is if it has an incentive; either what Sal suggests (the parents care for the incubator), or the ants that hwhatting mentioned (the parents are so rare that the carers have no choice but to raise someone else’s young to ensure the colony’s viability), or some other strategy. Otherwise, it’s a lot of trouble on the incubator’s part (and a lot of risk on the parents’ in transferring the embryo to a third party) for no reward at all.
EDIT: I should also say, it would also be useful to have the specific details of this reproductive strategy in order to answer the OP’s question about the appearance of the species’ genitalia.
The inseminator would have a penis, yes, while the ovipositor and the incubator don’t necessarily need anything different to a boring old vagina. If you’re open to the idea, the oviposition could simply take place in a manner similar to marsupial birth, where the fetus crawls out of the mother on its own into the pouch (or in this case, into the incubator’s womb/pouch), where it continues developing. This strategy is probably the least alien, most “mammalian” (in the sense of Terran mammals); you essentially get one ‘male’ and two ‘females’, both of whom give live birth. It seems likely the ovipositor would cooperate with the incubator in this strategy, nurturing and protecting it, while the inseminator would behave like a typical mammalian male, just out to fertilize as many eggs as he can.
On the other hand, if you’re really set on having the zygote oviposited into an incubator, in a similar fashion to insemination, then the mother might need a specialized eversible organ like Xephyr mentioned, so she can first receive the sperm and then impregnate the incubator with the zygote. This sounds like it could take place in a similar manner to traumatic insemination, where for example male bedbugs pierce the female’s body cavity and forcibly inject sperm, or like parasitic wasps depositing their offspring inside other animals.
Alternatively, fertilization could be external, so the mother deposits an egg, the father fertilizes it, and the incubator picks it up. This strategy reminds me of some frogs, who leave all their offspring with a single carer until they develop from tadpoles to froglets. Or the unfertilized egg and sperm could both be delivered directly to the incubator by a similar organ in both parents, an ovipositor/penis, in which case you essentially have two ‘male’ sexes, both of whose genetic material is necessary to create viable offspring, and one infertile ‘female’, in whom the egg and sperm meet and become viable. This strategy also sounds like parasitism, though I can’t think of anything similar in the real world.
EDIT2:
Pedant wrote:That said, we're all looking at a standardized third sex at the same stage of the life cycle. Is it possible, for example, to have an animal that is sexually active and capable of reproduction at one stage of its life cycle in a hermaphroditic manner, which then undergoes metamorphosis and becomes either male or female? The cultural implications (assuming that a species with this system achieved sapience) might be interesting...
Autoparasitism! Fun. Such a drastic metamorphosis doesn’t sound likely in a mammalian species, but it could perhaps be that when a female reaches a certain age, she undergoes menopause, and she becomes a target for younger females looking for a surrogate womb for their fetus to crawl into. Or, to put it less violently, young females are incapable of hosting a fetus, and older females undergo a sort of second puberty, in which their womb develops fully and they seek out younger females to surrogate for. True hermaphroditism and sex changes just don’t really sound like an option for a mammalian species.