What do I tell my clients?

Keep it generic and note that you won't be available by email or phone during that time.

What do you tell them when you go on vacation normally? I'd keep it generic.

Keep it generic, if they know you- they probably know its wedding related.

'I will be out of the office, returning on XXX. I will return your message after my return. For immediate assistance please contact xxxx.'

I'd also keep it generic and just state that you will be out of the office and unavailable.

Business email sites like outlook have a feature where you can create a ooo response for when people email you.

I think it would be best to defer to the etiquette of your industry, I don't feel there is official wedding etiquette on this issue. In some friendlier or quite personal business relationships it might be acceptable to share that detail - like in my role, for example, talking about a honeymoon would be encouraged! - but I'd think in most industries it would be best to be generic.

Set up an out of office with a POC while you are out.

I would advise against emailing that you'll be out for an extended period. It sounds like you have a health issue or something more serious than a honeymoon and will leave your clients wondering. Set an out of office message with your date of return and alternate contact for urgent requests. No need to explain why you're out, in my opinion.

Generic. They don't need to know why you are out of the office, just that you are. Let them know who they can contact in your absence. If no one, let them know when they can expect to hear back from you.