What is a proper gender-neutral form of himself or herself [duplicate]

Solution 1:

Singular they is a usage hallowed by time -- it's actually older than singular you -- and, since reflexive pronouns are inflected on both the pronoun root and the -self/-selves reflexive suffix (e.g, my[sg]-self[sg] vs our[pl]-selves[pl]), the correct reflexive for singular they is themself.

Dictionaries, by the way, are not reliable sources for grammar, just as grammars are not reliable sources for lexical meaning.

Solution 2:

The reflexive and emphatic pronoun is themselves. The OED has no entry for themself. An example of its use in the way you have in mind might be:

If a student needs to ask for permission to leave the campus during the week, they must do it themselves, and not ask a fellow student to do it on their behalf.