Dear god this was a rabbit hole...
The males wear armored cups meaning they are protecting something down there. Whether their piece is a red rocket, corkscrew, or a floppy sausage is up for debate but clearly they are armoring something down there. Otherwise seeing as how they can take heavy gunfire and be fine, I don't see why they would single that part out if they weren't protective of it.
We also know that they also wear some form of undergarments meaning they do have some level of modesty. How large those speedos are is up to the individual, but we know they wear them. So with that being said, its more likely they have an ass and some form or genitalia. Else if there was nothing, there is nothing to hide down there and they do armor themselves, they for sure armor the front and in some cases double up.
The mandibles also don't seem to be anywhere near a sexual organ as well. The mandibles from what we know are able to be prehensile stabbing tools as Fugitive could do so, and Upgrade (though a hybrid) was able to pry off a guy's skull with them. Meaning that they are likely first and foremost a method of capturing prey items. The mandibles also grow extra tusks as evident by City Hunter and Wolf, meaning that if used in the act of mating, not only should we be seeing more facial scaring even on the males (seeing as we have no reason to believe the females couldn't couldn't grow extra tusks) but we should also be seeing more breakages. If the mandibles are present to lock the heads in place, if one or both partners move a certain way they run the risk of snapping a mandible. Either at the tusk or potentially breaking a bone in the Yaujta's face. To that end and being the primary form of communicating I find it unlikely. The PredAlien's mandibles also extended forward to do that while all other Predators could bend. The Predator mandibles don't exactly extend so much as they flare outwards. They appear shorter by comparison because they are usually relaxed or tighter to the face. It was also the inner jaw depositing the embryos which is a Xenomorph biological item that we have no reason to believe Yaujta possess. The resulting offspring had also been Human derived Aliens and lacked Predator traits. Meaning that if it was Predator in nature we would be seeing more resulting traits like mandibles or even coloration. All Aliens spawning from Chet did not possess any Predator traits and thus we have no reason to believe that this is a function of Predator DNA. If nothing else it could've been caused by the mutation, however a mutation doesn't have to be from one or the other parties involved, it could just be a new sequence resulting in Egg vomiting.
Chet was also becoming a Queen during Requiem and while she didn't live long enough to do it, she was at an in-between stage between a Warrior/Drone and the Queen, meaning that this could also just as easily be a function of the Praetorian. But this trait is also heavily reliant on pregnant creatures being available to be a literal ugly mother f----- with. Meaning that in the absence of a pregnant individual we have no reason to believe Chet would've even bothered to utilize it, this trait is likely opportunistic. On the side of the Predators while it's not unheard of for someone to have sex with their wife, its another to supplant an individual embryo of another species with their own and to my knowledge there is no naturally occurring species that does this. That also calls into question where the Yaujta species even came from if they require a Yaujta to do this but you can't get one without another Yaujta and a pregnant female. Essentially you require a Yaujta and a pregnant female to get a Yaujta but you need the Yaujta in the first place to do this.
We also know that their mouths don't have cheeks and while the flaps between the mandibles would minimize the loss of semen to some degree, the Predator's would have to aim pretty well and that's also noting that there could be some substantial difference in the area of the mouth among individuals. If it's a proboscis as well, we have no evidence the Yaujta even possess one but more to that problem is that the mouth is the primary form breathing. The Yaujta if it was a protracted experience runs the risk of suffocation. The Helmets would also be adapted for this problem seeing as while some do have areas where the mouth is visible, the Yaujta still can't open up their mandibles all that wide to release some sort of proboscis nor could they properly shoot semen as the mandibles would be in the way. The mandible portion of the helmet also prevents them from locking up. The Helmet also is an effort to remove, though not a problem for Upgrade, the other Predators had to dislodge a few items before the helmet could be removed. While they could just remove it before the efforts begin, we also know by Predator 2 and by Requiem that it isn't uncommon for them to walk around with it on. Seeing as there could easily be females aboard any given ship it would make things unnecessarily harder for them to do the deed if they have to take a few moments to take the helmet off.
All of this is my long-winded rabbit hole of an explanation to say that Chet's vomiting is a mutation or Praetorian trait and that we have no reason to simply think that the Predator wouldn't just stick their sausage into it. It's just easier to assume that they have a penis than over-complicating it with genetic mutations, a creator of an entire species, and resulting in a creature that vaguely resembles them and requires a parasitic host to make more...why does this all sound familiar?
