No
Many years ago my wife's biology teacher made a simple statement: A species that cannot propagate/reproduce is not a species. Mules, for example, are not a species. This, despite the fact that they have genitalia. Mules can be either male or female but are born sterile. This might meet the intent of your question but does not meet the letter of your question. (Although I doubt that it even meets the intent of your question because mules are not on the list of animals that exhibit homosexual behavior, which one might conclude if the sterile nature of the animal combined with its genitalia would result in no practical preference for sexual activity.)
But herein lies the problem with your question. Humanity is a species. While we are learning today that the perception of sexuality and identity is more complex than our past has led us to believe, the simple fact remains. We are a species because we can reproduce. We can reproduce because the species has and depends on two biological sexes for reproduction: male (sperm donor) and female (egg donor).
But could there be such a person?
@NixonCranium points out ambiguous genitalia. The condition identifies, to put it simply, mishapen or misaligned genitalia. They are underdeveloped or the external organs don't match the internal organs (e.g., a penis and a uterus in the same body)—but they're there. So I believe it's use to identify the plausibility of your question is out of order. Similarly, there are other conditions, too, such as Vaginal Agenesis, which (simplistically) is the female born without or with underdeveloped interior sexual organs (e.g., uterus). Another is Cloacal Exstrophy which (very simplistically) is a disorder of the large intestine that compromises either penis or clitoris. But in all these cases, the child is (fundamentally, please don't argue with the simplicity of this statement), male or female. Which is shrouded by the conditions, but as genetic testing improves, so does the ability to identify reproductive gender.
I can't find (conveniently, I admit) a reference to an example of a birth where there were no genitalia or internal reproductive organs.
What all these conditions underscore is that being born without genitalia (not considering the other aspects of your question), as you are seeking, is a special (pronounced spee-see-al, not speh-shal) deformity. Humanity is meant to have genitalia. That's the "external" evidence that we're a species at all. In short, you're asking if humanity can breed, not just a mule, but a fully realized mule. A healthy human that cannot identify with any gender, any preference, or any identity.
As a final point before I move on, I also agree with @NixonCranium that even if a child was born without genitalia, all that's missing is the "external" evidence. Even if the child were born without genitalia and internal reproductive organs, genetically there are still chromosomal alignments. I also agree that the results are damaging (all these conditions have consequences great and small). As an example, think about running an automobile without an air filter, or brakes... The vehicle is meant to have those things but it can operate without them—with a high likelihood of consequence, great and small.
Therefore...
Therefore I must conclude that it is not plausible (and, I suspect, not at all possible) for someone to be born into the conditions you're asking about.
But should that stop you?
This depends on what the point of your efforts is. We sometimes depend too much on "reality." Does it matter if such a person can "really" exist? Given today's political and medical climate, I suspect that a well-written story centered on such a being would be very well received by one portion of society and absolutely hated by another. And whether or not it can "really happen" (assuming it could be proven to the level of the Hard-Science tag) wouldn't actually be that important to either side.
So, frankly, I'd ignore my answer and any other answer that doesn't provide the sense of assurance you're looking for. It's your world. Go tell your story.
