There are always some employees in all agencies who can't strike, either because management etc, those in different unions (but usually doing different work), and some are declared 'essential employees.'
But I warn: these are pretty minimal and usually the work they would be limited to providing is essential or emergency work. For passports, that might mean only those with documented proof of family emergencies or similar.
They might have resources to do simple stuff like give out already prepared passports - but stuff like that would likely be management view of what services they can provide with the minimal resources. (And important, that particular one is only my completely random guess as an example)
Point being, don't assume you'd be in the group that would get the emergency treatment - what government thinks is essential or urgent is going to be different than what you think.
