Their signature to say you will call 911 at their expense in any medical emergency and no one can rule that out with these two children, that way you are covered and you would have done the right thing anyway, prevention is always better than cure and him having his puffers to hand is definitely the right thing, but the ambulance and emergency for sure as back up.
After hospital the doctors would have ensured his correct meds (puffers) are in place and work well for him, got to feel sorry for any child with a breathing problem such as asthma though. Most of these things happen in mid to older age with smokers not a young innocent child.
Ensure the puffers are always well supplied and not empty of course.
If you aren't comfortable though remember you don't have to do it at all and can refuse care of the child.