Would use compromising as a conflict management style where by both i and the worker have important issues and we both need equal solution.

I would approach both the coworker first and try to explain to him that howni have covered for him severally so that we can collaborate together and come up with a solution that we will present to the manager that will favor both of us. for instance looking for another coworker to cover for the shift or work for extra hours to cover what would be required.
the possible outcome is that the coworker might not consider this important may be because he think the manager will consider his emergency more than my vacation.

And the manager might also refuse our solution as he can say that its possible to give me o nether chance for a vacation and the other coworker has to attend to his emergency.