Unlike the engineered hermaphroditic humans in Lois McMaster Bujold's Vorkosigan universe, who might be speaking a future version of English, your hermaphrodites are an alien species and do not speak English among themselves. They will therefore have a pronoun in their language that has no correspondence in contemporary English.
If you want your narrator to convey this linguistic feature to your readers, you may invent (part of) that alien language and use that alien pronoun in your narration. This should not be a problem to your readers after some introductory explanation (in text or in a preface) and a few reminders every now and then during the narrative.
If you want to write in English without using words from another language, you could look at what translators do when they translate text into English from one of the existing genderless languages.
In my opinion, the best you can do, if you want to represent the absence of gender in a gendered language such as contemporary English, is:
- use neuter pronouns (it/its/itself)
- use one gendered pronoun (either he/his or she/her) consistently throughout the text, and explain your usage either in text (through the narrator) or in an author's note prefacing the text
- use one
- avoid pronouns
he or she
I have frequently seen the second option employed in published fiction, often with female pronouns, often by feminist authors, and mostly for human or humanoid species. Sometimes this was done without the explanation, as a conscious play on gender expectations in the reader.
A recent close example is Ancillary Justice by Ann Leckie, in which English "she" is used for both genders in a culture that "doesn't much care about gender" and does not have gendered pronouns in its language:
She was probably male, to judge from the angular mazelike patterns quilting her shirt. I wasn't entirely certain. It wouldn't have mattered, if I had been in Radch space. Radchaai don't care much about gender, and the language they speak – my own first language [represented by English, in the book] – doesn't mark gender in any way.
The first option (using "it") would emphasize the fact that your beings aren't human, which might or might not be something you want. A clear advantage is that your readers wouldn't have to constantly remind themselves that "she" doesn't mean "female".
Singular they isn't commonly accepted as correct, so I don't see it as an option for most writers. As the English language changes, singular they may eventually become Standard English or even replace gendered pronouns, but today too many readers see it as grammatically incorrect or "policing speech". I wouldn't use singular they in my writing before it has been accepted by the majority of my target audience. Other writers may want to take a stand for gender neutral language and use singular they in their writing as a political act.
This is the option I prefer. It is free of prejudice, clearly singular, and its slight suggestion of Broken English fits the fact that it is spoken by aliens perfectly.
Certain languages have no third person pronouns and their speakers use nouns (such as boku "servant" in Japanese) to refer to other persons. Small children use their names to refer to themselves. Something akin to this will give a decidedly alien touch to your aliens' speech.
This is my second favourite option. It's not the first, because it will take quite an effort to worldbuild the aliens' culture to find a term that works well.