It's going to vary a bit by jurisdiction, everywhere handles things a little differently

The coroner's office should have an office number and you can certainly try calling that. It may or may not be staffed overnight or over the weekend and they'll have some sort of on-call procedures in place (in my county, when they don't have anyone in the office, their phones actually come through to us at the dispatch center to have the on-call coroner paged. Generally speaking we don't do that for the general public, just for police, hospitals, etc.)

Whatever funeral home you intend to use may also be able to handle it.

But in general, just call 911. I won't lie, a lot of what happens after that kind of happens in a black box from my perspective, I take the call, hang up and police/fire/EMS go out and do their thing and I get very little follow-up from there. But they have the experience with this kind of thing, they know what steps to take from there.

I also get a decent amount of calls where my callers are kind of clueless about what's going on, it's happened that they tell me the patient is conscious and alert only for the field units to report that they are in fact stiff and cold to the touch and an obvious class 5, and the opposite way around where they're sure someone is dead and when they get out there the person is in fact up and talking and seems to be in perfect health, and of course everything and anything in-between. So it never hurts to have someone go out there to make sure things are actually as they seem. And of course we want to double check to make sure there wasn't anything suspicious about the death as well.

I remember I had a caller one time who had been transferred to us from a nearby county where she was located. She told me her father had just died and she was having trouble getting ahold of her relatives in our county to let them know so she wanted us to go try to make contact with them for her (this would be about a priority 4 BTW, emergency and non-emergency calls all get handled through our central dispatch here)

Of course she didn't have her relatives addresses, good phone numbers or much of anything for us to actually help us make contact with her relatives. But I was trying my best trying to help her, asking a lot of questions trying to figure things out trying to get her to describe where they live etc.

But the more I'm talking to her, things just seem kind of off, so I ask her when exactly her dad died

It was like literally right before the call, she was still sitting around in the home with the body and the first thing she thought to do about it was call her relatives that she apparently barely spoke to anyway.

Which, fine, I get wanting to let your relatives know about a death in the family, and different families and cultures have their own funeral practices and such, but you probably want to do something about the corpse in your living room first.

So I got her back over with the dispatch for her county, both so they can do whatever they need to about notifying the coroner and whatever other policies they have in place and because her local police would probably be better able to run the information through their system to find contact info for the relatives than I would be over the phone with her.

Still 911 or 112 or your local equivalent. Then they'll decide. If they think a crime happened they want police there quickly. If they don't, they'll have someone come around when there's free time.