Yes, you can use selectionDay:
https://api.daypilot.org/daypilot-navigator-selectionday/
If you have a Navigator instance called "nav":
var nav = new DayPilot.Navigator("...");
// ...
you can read the value like this:
<input type=button id="email" onClick="location.href='email_event.php?date=' + nav.selectionDay" value='Email Event'>
It will be automatically converted to full ISO 8601 date/time format without timezone spec, e.g. "2020-04-17T00:00:00".
You can get custom text format using toString():
<input type=button id="email" onClick="location.href='email_event.php?date=' + nav.selectionDay.toString('yyyy-MM-dd')" value='Email Event'>
There are also two more properties which may have different values than selectionDay if the selectMode is set to "week" or "month":
https://api.daypilot.org/daypilot-navigator-selectionstart/
https://api.daypilot.org/daypilot-navigator-selectionend/