There appears to be an incompatibility between Appointments+ and WPMUdev's Domain Mapping plugin when SSL is involved (unless it's a configuration problem I haven't yet identified; I didn't set up the plugin in the first place, but I haven't yet found an obvious place to configure Appointment+'s behaviour in this respect.)
We run a WPMS instance with WPMUdev's Domain Mapping plugin. We force SSL as well as force the instance's original domain for all wp-admin functions (or else we'd get SSL CN (domain) mismatch).
When attempting to select a time slot in Appointments+, the browser attempts to POST an AJAX request to https://mapped-domain.com/wp-admin/admin-ajax.php which, of course, fails, because the CN of the SSL certificate is for original-domain.com.
I don't yet know why the plugin is using SSL for this request (mainly because I've never gotten past it, so I don't know what is expected to happen next). Maybe it's some sort of configuration issue (because I can't believe that Appointments+ is hard-coded to use SSL), or else Appointments+ is being clever, but not quite clever enough.
Appointments+ needs to be cognisant that, if it is running in a WPMS situation with Domain Mapping, it needs to use the original domain for SSL requests.