There is no need for a phrase like English “his/her” in Spanish, because there is only one third person singular possessive pronoun in Spanish, su, and it is not marked for gender.
If the need arises to make it clear that something belongs to “him” or to “her”, then there is the alternative of rephrasing using de él or de ella. In this case the noun must be preceded by the definite article, which does of course have gender, but it's the gender of the possessed noun, not of the possessor.
- El auto de María = el auto de ella = su auto
- El auto de José = el auto de él = su auto
- La casa de María = la casa de ella = su casa
- La casa de José = la casa de él = su casa
With the third person pronouns the situation is reversed because both the subject and the direct object pronouns have gender, and there is no simple way to include both except by writing él/ella (subject) or lo/la (direct object). The indirect object pronoun is gender-invariable le.
When the sentence has a direct object it is impossible to avoid this kind of double pronoun. When the problematic pronoun is the subject, though, it is often easier because Spanish can usually simply drop the subject pronoun, so there is no need to mention either él or ella. There is no equivalent to the English usage of “they/them” as gender-indifferent pronoun.