What's your preference?
Everyone must consult his or her manager is clumsy, but it does avoid 'his manager' which is clearly unacceptable - unless everyone on this occasion just happens to be male.
Everyone must consult their manager' uses 'their' with the singular 'everyone'. Is that okay? Yes. It's not only a necessary device in modern business writing, but fully justified by usage and even officially approved.
The three major dictionaries - the Oxford English, Websters and the Dictionary of English Language all just note that plurals such as their, they, and themselves are used with singulars such as everyone, every or any. There's no sense that the dictionary editors are wagging their fingers.
Henry Fowler's advice
That great guru of English usage Henry Fowler provides the example '...anybody can see for themselves' and describes it as 'the popular solution'.
You know pedants who say it's ungrammatical?
Refer them to the dictionaries and Fowler. Pedants take notice of such authorities, but left unchecked, pedants stifle the language.
'Somebody parked in the wrong place, didn't they?'
Relax. Even Henry says it's okay.
Thanks to Clarity and the late Professor Robert Eagleson and his task force colleagues.