Email is one of the top tools for business communication. People expect swift responses to their requests. If this expectation will not be met because you are away from your desk, your client or colleague needs to know.
The Management Office will be closed on [date] for [Holiday Name]. Any inconvenience caused is much regretted.

Hey, Thanks for your email. I’m not in the office and am on a family vacation. I assure you that all your emails will be answered as soon as I return to the office on [date]. Kind regards.
What we need in our work communication is not more professional politeness or less formal, chat-based messaging applications like Slack. We need honesty. The problem is that we’ve conditioned ourselves to see honesty as self-indulgent or disrespectful. I’d argue the opposite is true. Honesty, even if it’s a bit more inconvenient for all parties in the moment, pays dividends later. It builds trust. When my partner Anne Helen Petersen and I were interviewing people for our forthcoming book on remote work, a frequent lament from both middle managers and workers was that they didn’t feel like they knew how to succeed in their jobs; that they were guessing what their superiors and coworkers wanted and, even when they asked, they didn’t quite trust the responses they got back.