Is it possible for a manager to receive an email when a player changes availability?
No, it is not possible to receive an email when a player changes availability.
An alternative would be to set an Availability cutoff time for your games and events. Any player wanting to make a change after your cutoff time would have to contact you directly.
Go to your Availability tab > Availability Preferences
You can also set a Manager Notification in your team Preferences tab > Notifications to have TeamSnap send you a list of available players prior to games and events