IT manager of more than a decade here. Vacation days are earned. They're part of your pay. If you're not taking vacation days, you may as well be not cashing a paycheck. That's unacceptable.
If several folks request off the same day I'll gently ask if anyone is willing to swap. If they're not willing or able, it's OK. I'm paid more and I will work the overtime needed to cover them, because they earned it and it's their time. Period. The company will survive a few days of less stellar work so that my guys and gals can live their life and work a job where they feel confident that they're taken care of.
So glad my work has sick days too. Mental health is a good enough reason to use one and you don’t even need to use a reason. I do my best to show up for important things but I am so thankful I can call in sick or use a vacation day any time I want for the most part….