Why is it more likely you'll have internet access when you don't have a phone? If you happen to find a computer, what's insufficient about writing an email?
Why hide emergency contact info? Put it in Emergency group on everyone's phone.
The goal should be to send a single message with nature of emergency, what you need from them, and what they do like call police
I could see a private reference page with important info. Including checklist for various scenarios. That's what I thought you did. Then print out a copy and put in binder somewhere at home.