No plan but should. I've been thinking about it. Put a sign on the front door maybe? I'd need something similar in the truck. . .
The habit of police departments and, perhaps to a lesser extent, of EMS personnel, is to leave doors open when they respond to an emergency at a home. When I could see that someone had a pet, I would often have to remind people to close the door. If someone is going to the hospital, or worse, the concern is going to be for that person, and relatives might not even think of the pet until much later, particularly if the pet is outdoors and afraid to come back in because of all the strangers and upheaval going on there.
I bought these keychains and wallet cards on Amazon: https://www.amazon.com/gp/product/B0817FFB3L/ref=ppx_yo_dt_b_search_asin_title?ie=UTF8&th=1 I also keep all the emergency info for my pets written out and on my refrigerator. Emergency response people know to check your refrigerator for emergency information.
My pets are my biggest concern whenever something happens. Two years ago I had a fall and had to take an ambulance ride. First there was a visit from a home health service and another one from an in-home X-ray service. All this because I was too stubborn to admit a bone was broken. They all propped the door open to bring in their equipment. In my case my daughters were there to put the cats in another room but if not I would have been frantic that they would get out.
Unfortunately, that would be a common problem, and it would probably keep me from calling an ambulance unless there were no other way for me to get to the hospital. Whoever it is that comes to your house in response to an emergency, police, fire, or EMS, they are likely to be focused solely on their job and, you're right, if they have to get a stretcher (gurney) or other equipment into your house, they are likely to leave a door open, and, in my experience, the police are likely to that even if they're not bringing anything in. Generally, dogs are more visible than cats are, so they're more likely to be noticed and shut up in another room, in part because people view them as a biting thread. Cats are more likely to hide, but they might come out from hiding to make a rush for an open door in order to escape the strangers and commotion going on in the house. As a paramedic, I would usually ask about pets, and check on the door from time to time, but not everyone thinks of it. Yes, if someone is having a medical emergency, that emergency should take priority, but usually, that can be done without jeopardizing the pets. I know that I wouldn't be at all pleased if I went by ambulance to the hospital and returned to find that my cats were missing because they were allowed to escape the house at what would have been a scary time for them.
Make friends, family, and neighbors aware of any animals you have. If they know about the pets, they can perhaps check on them if you are missing for any period.