Here's the reason the ASM should have this info.
You are calling the show.
An actor breaks a leg (or gets sick, or hurt) and goes to the hospital.
Have you ever tried to call 911 or someones emergency contact info while calling a show and trying to make decisions on how to keep the show going without stopping it. Being an SM who has had to call 911 (three times) while calling the show, it's nice that someone else has the info.
Also, I feel strongly the ASM should have a copy of the calling script. (Ever need to rush to the bathroom, loose all headset communication, have to call 911, put a fire out in the booth . . .) Nice to know someone else can carry on.
I agree with Matthew 100%. Medical forms are Continental and for the necessary personnel, and that includes the entire stage management team. A prompt book should never (in my opinion) be just left laying around with that information in it anyway.
As someone who's been in similar situations (gotten sick in the booth, had an op get sick in the booth, had to call 911...though that was backstage...it was good to know that my team and I were on the same page and things could run smoothly.
Also, I believe that if an injury happens during the course of the show, while I am ultimately in charge, it is my ASM's job to manage the situation backstage. I will make the decision through communicating with them whether the show stops or not, but they are going to be that actor's first line of assistance and they need to know if they're allergic to penicillin or not.
Also, if EMS does come, it's great to be able to just rattle off the pertinent info without having it relayed through a headset.