I've had several people ask, and now I can finally say - Yes! There is a multi-user version of Asthma Tracker & Log!
You can export your database from your existing version and import it into the multi-user version so you can keep all of your data.
Asthma Tracker & Log Multi User can be found in Google Play at: https://play.google.com/store/apps/details?id=rovingreptiles.asthmatrackermulti and will be submitted for approval to the Amazon App Store shortly.
Thanks for all of your feedback, please let me know if you run into any issues with this version!