To avoid using gender-specific pronouns "his/her" you can use "their", but it doesn't make it plural.
Regarding the title of your question, you need a singular pronoun for "each employee" because you consider them individually and not as a whole.
I think in your sentence you should use "any" instead of "each" because they could be from zero to all the employees who may have such a request and you don't care who or how many they are.
"Each" is used when you want to say that an action is performed individually by the whole members of a group.
each employ should focus on their (his/her) task