I have always had a "disaster recovery plan" in place, so since Day 1 I've known exactly who would be taking care of what, in the event I was incapacitated (long- or short-term).
When I was a reseller and doing everything myself, I had a file with my provider's contact information and the basic starter URLs to my billing system, helpdesk, et cetera. My provider, in turn, had a piece of paper stating that if ever a certain individual contacted them, they were authorized to work with that person to support my clients for either a period of one week, until I got in touch with them, or until that named contact person was able to provide a certified death certificate and proof of legal transfer of the company.
When I moved to dedicated servers, I contacted the prospective provider and asked if they would be willing to work with me in such a plan: I'd provide them with the name of the authorized contact person, and if this contact person did contact them, they'd work with that person, et cetera. I had to beef up my information-sheet a bit, because running a server takes a bit more information than a reseller account; but it only took me about 30 minutes to update everything.
This setup gave a smooth transition in terms of someone taking care of my clients' support requests, someone making sure the back end was running, and legal protection for the provider: they had legal proof of who was authorized, in what circumstances, and for what scope of authority. It also listed the circumstances under which "control" would be passed back to me, so again, the provider knew what was going on. They weren't required to make any judgment calls or guess, or take any actions that could put them at risk. Plus my clients were taken care of, and wouldn't even have to know that I was, say, in the hospital having an emergency appendectomy. All they knew was the server was running, their support tickets were being answered, everything was status quo.