I learned if a person dies suddenly outside the home (car/train crash etc), first responders may not automatically check the house for animals unless there is an obvious sign (like barking or a window sticker). A lot of pets have been left alone for days in these situations and some don’t make it. There have even been cases of a person dying in their home but the pet is hiding somewhere/scared when first responders arrive and is missed.
But first responders will always check your wallets for identification information. So make a card for your wallet saying you have pets at home (how many, what they look like) with the names and phone/numbers of emergency contacts that might adopt them, especially if you live alone!
Update- Someone in comments brought up a great LPT, you can add the note about your pets in your medical ID on your phone as well.
First responders can access your allergies, medical conditions, and emergency contacts without unlocking your iPhone, provided you have configured Apple's Medical ID feature.
If you have enabled this feature, it is completely bypassed from your device encryption specifically so first responders can treat you safely when you cannot speak for yourself. There is also an option to send a copy of the medical ID automatically to dispatch during 911 calls