I am wondering how to use a grammatical gender in Italian for non-gender-conforming people. For example, how should I use adjectives when referring to a person who does not identify as either male or female?
Italian has just two grammatical genders, masculine and feminine (no “it” or the like). Every noun (including objects and abstract notions) has one of these two grammatical genders, which, even in the case of humans or animals, doesn't necessarily relate with the sex (if any) of the person or animal involved.
For instance, sedia (chair) is feminine, while tavolo (table) is masculine; sole (sun) is masculine, while luna (moon) is feminine. Many animals have a single word for their kind, either masculine or feminine, independently of the actual sex of a single individual. For instance, pesce (fish) or topo (mouse) are masculine, and you should resort to a periphrasis to specify that a particular fish or mouse is a female; the reverse holds for tigre (tiger) or tartaruga (turtle), which are feminine.
For human beings, we normally use distinct words for females and males (or equal words with adjectives and pronouns agreeing with different genders): scrittore ([male] writer) - scrittrice ([female] writer); professore ([male] professor) - professoressa ([female] professor). But even so, some generic terms that refers to generic humans have a specific grammatical gender (an Italian noun cannot not have a gender). For instance, essere umano (human being) is grammatically masculine, while persona (person) is feminine. Moreover, such nouns as spia (spy) or guardia (guard) are feminine, whatever the person they refer to.
All of the above to say that Italian language and its speakers perceive grammatical gender as less connected to personal gender (or lack thereof) than, say, in English, where “he” and “she” refer just to male and, respectively, female people (or some animals and rare exceptions, such as ships).
(Once we know the grammatical gender of a noun, the adjectives and pronouns that refer to it have automatically the same grammatical gender: Mario è una persona molto generosa = “Mario is a very generous person”.)
So, I believe that a person who doesn't recognise themself as either male or female can use whatever words they prefer, perhaps choosing a single conventional grammatical gender, or alternating both or whatever. Keep in mind that in Italian no one objects if virile James Bond is called a spia (feminine only word for “spy”) or a woman singing as a soprano is called, well, soprano (a masculine noun in Italian).